Find a matrix $$X$$ such that $$2A+B+X=O,$$ where   $$A=\begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix}$$    $$B=\begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix}$$

Write the inverse of the matrix  $$\begin{bmatrix} \cos \theta  & \sin \theta  \\ -\sin \theta  & \cos \theta  \end{bmatrix}$$

A is $$2\times 2$$ matrix such that $$AB=BA$$ when ever B is $$2\times 2$$ matrix. Prove that A must be of the form kI, where 'k' is real number and I is the identify matrix.

Let $$A=\begin{bmatrix} 1 & -2  &  3 \\ 3 & 2  &  -1 \end{bmatrix}$$ and   $$B=\begin{bmatrix} 2 & 3 \\ -1 & 2 \\ 4 & -5 \end{bmatrix}$$ . Find AB and BA and show that $$AB\neq BA.$$

Given that $$\begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix},$$ if  $$A = \begin{bmatrix} a & b & c \end{bmatrix}.$$ Find $$a+b+c.$$

Compute the following:
(i) $$\displaystyle \begin{bmatrix} a & b \\ -b & a \end{bmatrix}+\begin{bmatrix} a & b \\ b & a \end{bmatrix}$$
(ii)  $$\displaystyle \begin{bmatrix} { a }^{ 2 }+{ b }^{ 2 } & { b }^{ 2 }+{ c }^{ 2 } \\ { a }^{ 2 }+{ c }^{ 2 } & { a }^{ 2 }+{ b }^{ 2 } \end{bmatrix}+\begin{bmatrix} 2ab & 2bc \\ -2ac & -2ab \end{bmatrix}$$
(iii)  $$\displaystyle \left[ \begin{matrix} -1 \\ 8 \\ 2 \end{matrix}\begin{matrix} 4 \\ 5 \\ 8 \end{matrix}\begin{matrix} -6 \\ 16 \\ 5 \end{matrix} \right] +\left[ \begin{matrix} 12 \\ 8 \\ 3 \end{matrix}\begin{matrix} 7 \\ 0 \\ 2 \end{matrix}\begin{matrix} 6 \\ 5 \\ 4 \end{matrix} \right]$$
(iv) $$\displaystyle \begin{bmatrix} { \cos }^{ 2 }x & { \sin }^{ 2 }x \\ { \sin }^{ 2 }x & { \cos }^{ 2 }x \end{bmatrix}+\begin{bmatrix} { \sin }^{ 2 }x & { \cos }^{ 2 }x \\ { \cos }^{ 2 }x & { \sin }^{ 2 }x \end{bmatrix}$$

Solve the equation for $$x,y,z$$ and $$t$$ if $$\displaystyle 2\left[ \begin{matrix} x & z \\ y & t \end{matrix} \right] +3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}=3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$$

If  $$\displaystyle x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right]$$
Find values of $$x$$ and $$y$$ .

Find $$X$$ , if  $$\displaystyle Y=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$$ and  $$\displaystyle 2X+Y=\begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}$$

Find $$X$$ and $$Y$$ , if  $$\displaystyle 2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix}+\begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix}=\begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}$$

Simplify:  $$\displaystyle \cos { \theta  } \begin{bmatrix} \cos { \theta  }  & \sin { \theta  }  \\ -\sin { \theta  }  & \cos { \theta  }  \end{bmatrix}+\sin { \theta  } \begin{bmatrix} \sin { \theta  }  & -\cos { \theta  }  \\ \cos { \theta  }  & \sin { \theta  }  \end{bmatrix}$$

Given matrix $$A = \begin{bmatrix}4\sin 30^{\circ} & \cos 0^{\circ} \\ \cos 0^{\circ} & 4\sin 30^{\circ} \end{bmatrix}$$ and $$B = \begin{bmatrix}4\\5  \end{bmatrix}$$ If $$AX = B$$
Write the order the matrix $$X$$

Determine whether given matrices product is defined or not. If the product is defined, state the dimension of the product matrix.

$$A_{2 \times 5}$$ and $$B_{5 \times 4}$$

If $$A$$ is a matrix of order $$(m\times n)$$ and $$B$$ is a matrix of order $$(n\times p)$$ then order of $$AB$$ is ................

For the matrices $$A$$ and $$B$$ , the product $$AB$$ exists but $$BA$$ does not exist. What can you say about the order of $$A$$ and $$B$$ ?

If $$A$$ is any square matrix then $$(A-A')$$ is always ______

If $$A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ , find $$A\cdot { A }^{ \prime  }$$ .

If $$A=\begin{bmatrix}1&2&3\\3&2&1\end{bmatrix}$$ and $$B=\begin{bmatrix}3&2&1\\1&2&3\end{bmatrix}$$ , find $$3B-2A$$ .

If $$A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ , then find $$A'A$$ .

If $$[x - 3] \begin{bmatrix}2x\\ 6\end{bmatrix}= 0$$ , then find the value of x.

Solve $$\bigl[\begin{matrix} x& 1\end{matrix}\bigr] \bigl[\begin{matrix} 1& 0 \\ -2 & -3\end{matrix}\bigr] \bigl[\begin{matrix}x  \\ y \end{matrix}\bigr] = (0).$$

If $$A=\begin{bmatrix} 1 & 3\\ 3 & 4\end{bmatrix}$$ and $$B=\begin{bmatrix} -2 & 1\\ -3 & 2\end{bmatrix}$$ and $$5A^2-B^2=5C$$ . Find matrix C where C is a $$2$$ by $$2$$ matrix.

By row transformation find $$A^{-1}$$ if:
$$A = \begin{bmatrix} 2 &1  \\ 4 & 2  \end{bmatrix}$$

If $$A=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix};B=\begin{bmatrix} a & c \\ b & d \end{bmatrix}$$ , then show that $${ \left( A.B \right)  }^{ -1 }={ B }^{ -1 }.{ A }^{ -1 }$$

If 2X - Y =  $$\begin{bmatrix}3 & -3 & 0\\ 3 & 3 & 2 \end{bmatrix}$$ and 2Y + X  =  $$\begin{bmatrix}4 & 1 & 5\\ -1 & 4 & -4 \end{bmatrix}$$
then find X and Y

If  $$A=\begin{bmatrix}2&-1\\ 3&2\end{bmatrix}$$ , $$B=\begin{bmatrix}0&4\\ -1&7\end{bmatrix}$$ . Find $$3A^2 - 2B + 1$$ .

If A = $$\begin{bmatrix}1 & 3\\ 3 & 2\\ 2 &5 \end{bmatrix}$$ , B =  $$\begin{bmatrix}-1 & -2\\ 0 & 5\\ 3 & 1 \end{bmatrix}$$ .Find the matrices D such that $$A + B - D = O$$   i. e. zero matrices 

Solve for $$x$$ and $$y$$ :

$$2\begin{bmatrix} x & 7 \\ 9 & y-5 \end{bmatrix}+\begin{bmatrix} 6 & -7 \\ 4 & 5 \end{bmatrix}=\begin{bmatrix} 10 & 7 \\ 22 & 15 \end{bmatrix}$$

If $$\begin{bmatrix} x-y & z \\ 2x-y & w \end{bmatrix} =\begin{bmatrix} -1 & 4 \\ 0 & 5 \end{bmatrix}$$ , then find the value of $$x+y$$ .

If A = $$\begin{bmatrix}0 & 1 & 2\\ 2 & 3 & 4\\ 4 & 5 & 6 \end{bmatrix}$$ , B = $$\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$$  
Find 3A - 4B 

Can the following two matrices  be added 
$$\begin{bmatrix}1 & 2 & 3\\ 3 & 4 & 5\\  5& 6 &7 \end{bmatrix}$$ and  $$\begin{bmatrix}4 & 3 \\ 5 & 6 \\ 9 & 8  \end{bmatrix}$$

If $$A$$ is $$n$$ squared matrix then $$AA'$$ and $$A'A$$ are symmetric.

If $$A= \left [ \begin{array}{c}4\hspace{0.5cm}3\\2\hspace{0.5cm}5\end{array}\right ],$$ find $$x + y$$ such that $$A^2-xA+yI=O.$$

By using elementary transformation find the inverce of $$A=\left[ \begin{matrix} 1 \\ 2\end{matrix}\begin{matrix} 2 \\ -1 \end{matrix} \right]$$ .

If $$A=\left[ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right]$$ , then $$AA=$$

Find the value of $$a$$ and $$b$$ if $$\begin{bmatrix} a+3 & b^{ 2 }+2 \\ 0 & -6 \end{bmatrix}=\begin{bmatrix} 2a+1 & 3b \\ 0 & b^ 2-5b \end{bmatrix}$$ .

By using elementary transformation find the inverse of $$A=\left[ \begin{matrix} 1 \\ 2\end{matrix}\begin{matrix} 2 \\ -1 \end{matrix} \right]$$ .

Write the following as a single matrix $$\begin{bmatrix} -1 & 2 \\ 1 & -2 \\ 3 & -1 \end{bmatrix}+\begin{bmatrix} 0 & 1 \\ -1 & 0 \\ -2 & 1 \end{bmatrix}$$

$$A$$ and $$B$$ are two matrices of same order $$3 \times3$$ , where $$A=\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 8 \end{bmatrix}$$ and $$B=\begin{bmatrix} 3 & 2 & 5 \\ 2 & 3 & 8 \\ 7 & 2 & 9 \end{bmatrix}4$$ . Find AB

If $${A_x} = \left[ {\begin{array}{*{20}{c}}{\cos x}&{\sin x}\\{ - \sin x}&{\cos x}\end{array}} \right]$$ and $${A_y} = \left[ {\begin{array}{*{20}{c}}{{\mathop{\rm cosy}\nolimits} }&{\sin y}\\{ - \sin y}&{\cos y}\end{array}} \right]$$  

Show that $${A_x}{A_y} = {A_{x + y}}$$ .

$$\left[ \begin {array}{11} 2 & 1\\ 1  & 0 \\ -3 & 4\\ \end{array} \right]$$ + $$\left[ \begin{array}{111} -1 & -8  \\ 1 & -2  \\ 9 & 22 \\ \end{array} \right ]$$

Determine the product $$\begin{bmatrix}-4 & 4 & 4\\ -7 & 1 & 3\\ 5 & -3 & -1\end{bmatrix} \begin{bmatrix}1 & -1 & 1\\ 1 & -2 & -2\\ 2 & 1 & 3\end{bmatrix}$$ .

If $$x+y=\begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix}$$ and $$x-y=\begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}$$ Find the value of $$x$$ and $$y$$

Find the matrix $$A$$ such that $$\left[ {\begin{array}{*{20}{c}}2&{ - 1}\\1&0\\{ - 3}&4\end{array}} \right]A = \left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 8}\\1&{ - 2}\\9&{22}\end{array}} \right]$$ .

If $$A=\begin{bmatrix}3&4\\7&2\end {bmatrix}B=\begin{bmatrix}7&4\\0&9\end {bmatrix}$$ Then $$AB$$

Determine the product $$\begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}\begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$$  

$$\left[ {\begin{array}{*{20}{c}}1&3\\0&0\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\, = \left[ {\begin{array}{*{20}{c}}x\\0\end{array}} \right]$$ , find the value of $$x$$

Compute $$\begin{bmatrix} \cos ^{ 2 }{ x }  & \sin ^{ 2 }{ x }  \\ \sin ^{ 2 }{ x }  & \cos ^{ 2 }{ x }  \end{bmatrix}+\begin{bmatrix} \sin ^{ 2 }{ x }  & \cos ^{ 2 }{ x }  \\ \cos ^{ 2 }{ x }  & \sin ^{ 2 }{ x }  \end{bmatrix}$$

Let $$A=\begin{bmatrix} i\ 0\\0\ i\end{bmatrix}$$ , then find $$A^4$$ . Where $$i=\sqrt{-1}$$ .

If A = -1, 1      
   A = {-1, 1}
   A = {-1, 1}
  A - {-1, 1} 
Find $$A \times A \times A$$

If  $$\left[ { \begin{array} { *{ 20 }{ c } }{ a+4 } & { 3b } \\ 8 & { -6 } \end{array} } \right] =\left[ { \begin{array} { *{ 20 }{ c } }{ 2a+2 } & { b+2 } \\ 8 & { a-8b } \end{array} } \right]$$ then find the value of $$(a-2b)$$ .

If $$A=\begin{pmatrix} a & 1 \\ 0 & a \end{pmatrix}$$ , then $$A^2$$ equals to-

Solve: $$f(x) = \begin{bmatrix}\cos x & -\sin x & 0\\ \sin x & \cos x & 0\\ 0 & 0 & 1\end{bmatrix}$$ show that $$f(x) f(y) = f(x + y)$$ .

If $$x+y=\begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix}$$ and $$x-y=\begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}$$
find the value of $$x$$ and $$y$$ .

Find the inverse of the following matrices by the adjoint method 

$$\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix} \\ $$

$$\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}$$

$$\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}$$

$$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$$

If $$A = \left[ {\begin{array}{*{20}{c}}i & 0\\0 & i\end{array}} \right]$$ , find $$A^2$$ .

If $$A = \left[ {\begin{array}{*{20}{c}}4&3\\1&2\end{array}} \right]\,\,\,\,\,\,\,\,\,B = \left[ {\begin{array}{*{20}{c}}2&1\\1&2\end{array}} \right]$$ Verify  $${\left( {AB} \right)^\prime } = {B^\prime }A'$$

If $$A=\left[ \begin{matrix} -2 & 3 \\ 4 & 5 \end{matrix} \right]$$ and $$B=\left[ \begin{matrix} 5 & 2 \\ -7 & 3 \end{matrix} \right]$$ , find the matrix $$C$$ such that $$A+B-C=0$$

Determine the product $$\begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}  \, \begin{bmatrix} 1 & -1  & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$$ .

Find the value of $$y-x$$ from the following equation.
$$2\begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix}+\begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix}=\begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}$$

By using the elementary transformation, find the inverse of the matrix $$A=\left[ \begin{matrix} 1 & -2 \\ 2 & 1 \end{matrix} \right]$$ .

Given: $$3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}$$ , find the values of $$x,y,z$$ and $$w$$ .

If $$A=\left[ \begin{matrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{matrix} \right]$$ find $$(A')^{-1}$$

Simplify:
$$\cos\theta.\left[ \begin{matrix} \cos { \theta  }  & \sin { \theta  }  \\ -\sin { \theta  }  & \cos { \theta  }  \end{matrix} \right] +\sin\theta.\left[ \begin{matrix} \sin { \theta  }  & -\cos { \theta  }  \\ \cos { \theta  }  & \sin { \theta  }  \end{matrix} \right]$$  

Find the matrix $$A$$ such that $$\begin{bmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{bmatrix}A=\begin{bmatrix} -1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15 \end{bmatrix}$$

Find $$AB$$ , if $$A=\begin{bmatrix} 1 & 2 & 3 \\ 1 & -2 & 3 \end{bmatrix}$$ and $$B=\begin{bmatrix} 1 & -1 \\ 1 & 2 \\ 1 & -2 \end{bmatrix}$$  
Examine whether $$AB$$ has inverse or not.

Prove that adjoint of a symmetric matrix is also a symmetric matrix. 

If $$A = \left[ \begin{array} { c c c } { 1 } & { - 2 } & { 0 } \\ { 2 } & { 1 } & { 3 } \\ { 0 } & { - 2 } & { 1 } \end{array} \right]$$ ,then $$A ^ { - 1 } =$$

If $$A=[2\  0\  1],$$ then rank of $$AA^T$$ is

Find  the inverse of matrix by elementary row transformations.
$$A = \left[ \begin{array} { r r r } { 1 } & { 2 } & { 1 } \\ { - 1 } & { 0 } & { 2 } \\ { 2 } & { 1 } & { - 3 } \end{array} \right]$$

If $$2\left[ \begin{matrix} x & 5 \\ 7 & y-3 \end{matrix} \right] +\left[ \begin{matrix} 3 & 4 \\ 1 & 2 \end{matrix} \right] =\left[ \begin{matrix} 7 & 14 \\ 15 & 14 \end{matrix} \right]$$ , find the value of $$x$$ and $$y$$ .

Given, $$A=\begin{bmatrix} 1 & 3 & 5 \\ -1  & 0 & 2\\0 & 4 & -3 \end{bmatrix}$$ and $$B=\begin{bmatrix} 0 & 3  \\ -2  & 0 \\0 & -4 \end{bmatrix}$$ Find $$A+B$$

If $$A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix},$$ such that $$AX=I$$ , then find $$X$$ .

Let E = $$\left[ \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} \right] .\quad F=\left[ \begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} \right] \quad and\quad x \in  R,$$ then find $$E+F$$ .

$$A=\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}, B=\begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix}$$
Find $$AB$$

If $$x =$$ $$\begin{bmatrix}4&1\\-1&2\end{bmatrix}$$ , show that $$6x - x^2 = 9I$$ , where $$I$$ is the units matrix.

$$A=|_{ 1 }^{ 1 }\quad _{ 3 }^{ 3 }|\quad$$ and $$B=|_{ 1 }^{ -1 }\quad _{ -1 }^{ 1 }|\quad$$ . Find AB.

If A = $$\begin{bmatrix} 4 & 8 \\ -2 & -4 \end{bmatrix}$$ find $${ A }^{ 2 }$$

$$A=\begin{bmatrix} 1 & 8 \\ 9 & 4 \end{bmatrix}B=\begin{bmatrix} 6 & 7 \\ 3 & 5 \end{bmatrix}$$ Find $$A+B$$

Find the value of 'x' and 'y' if :
$$2\begin{bmatrix} x & 7 \\ 9y & 5 \end{bmatrix}+\begin{bmatrix} 6 & -7 \\ 4 & 5 \end{bmatrix}=\begin{bmatrix} 10 & 7 \\ 22 & 15 \end{bmatrix}$$

If $$A=\begin{bmatrix} 2 & 4 \\ k & -2 \end{bmatrix}$$ and $$A^2=O$$ , then find the value of k.

If $$A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}B=\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$$ then $$A+B$$

If $$A = \begin{bmatrix}4 & 3\\ 1 & 2\end{bmatrix}$$ and $$B = \begin{bmatrix}-4\\ 3 \end{bmatrix}$$ , then find $$AB$$ .

$$A=\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}, B=\begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix}$$
Find $$BA$$

If $$A=\begin{pmatrix} 3 & 2 \\ 4 & 0 \end{pmatrix}$$ and $$B=\begin{pmatrix} 3 & 0 \\ 3 & 2 \end{pmatrix}$$ then, find $$AB\, and\, BA$$ .

If matrix $$A = \begin{bmatrix}2 & -2\\ -2 & 2\end{bmatrix}$$ and $$A^{2} = pA$$ , then write the value of $$p$$ .

Find $${A}^{2}$$ if  $$A= \begin{bmatrix} 1 & 2 & 5 \\ 3 & 4 & 1 \\ 1 & -1 & 2 \end{bmatrix}$$

If $$A$$ is symmetric matrix and $$n\in N$$ , write whether $$A^{n}$$ is symmetric or skew-symmetric or neither of these two.

Evaluate the following :

$$\begin{bmatrix} 1 & 2 & 3 \\ -1 & 4 & 5 \\ 0 & 3 & 6 \end{bmatrix}\begin{bmatrix} -4 & -3 & -2 \\ -6 & -1 & 0 \\ -5 & -2 & 1 \end{bmatrix}$$

Find $$A+B$$

$$A=\begin{bmatrix} 1 & 3 & -1 \\ 2 & -5 & 4 \\ 3 & 2 & -6 \end{bmatrix}$$ and $$B=\begin{bmatrix} 2 & 4 & 0 \\ 3 & -4 & 5 \\ 2 & 3 & -5 \end{bmatrix}$$

$$A=\begin{bmatrix}3&5\\2&2\end{bmatrix}$$

$$A^2=?$$

Show by an example that for $$2\times 2$$ matrices $$A \neq O$$ and $$B \neq O, AB=O$$ .

If $$A = \begin{bmatrix}ab & b^{2}\\ -a^{2} & -ab\end{bmatrix}$$ , then show that $$A^{2} = 0$$ .

Using elementary transformations, find the inverse of matrix, $$A=\begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix}$$ .

Multiply the given matrix:

$$\begin{bmatrix}a & b\\ -b & a\end{bmatrix}\begin{bmatrix}a & -b\\ b & a\end{bmatrix}$$

If matrix $$A=\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$$ , write $$AA'$$ .

If $$A=\begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix}$$ , show that $$A^{2}=O$$ .

Perform the given operation: $$\begin{bmatrix} a& b\end{bmatrix} \begin{bmatrix}c\\ d \end{bmatrix} + \begin{bmatrix}a & b & c & d\end{bmatrix} \begin{bmatrix}a\\ b\\ c\\ d \end{bmatrix}$$

Find the inverse of the following matrix by using elementary row transformation :
$$\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$$

Addition of matrices defined if order of the matrices is _______ .

If $$A\quad =\quad \left[ \begin{matrix} 2 \\ 5 \end{matrix} \right] ,\quad B\quad =\quad \left[ \begin{matrix} 1 \\ 4 \end{matrix} \right] \quad and\quad C\quad =\quad \left[ \begin{matrix} 6 \\ -2 \end{matrix} \right]$$
$$A-B+C$$

Given $$A\quad =\begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}\quad and\quad B\quad =\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}$$ Find
$$AB$$

Let $$A=\begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix},B=\begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix},C=\begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$$
Find the following:
$$A+B$$

If $$A\quad =\quad \left[ \begin{matrix} 2 \\ 5 \end{matrix} \right] ,\quad B\quad =\quad \left[ \begin{matrix} 1 \\ 4 \end{matrix} \right] \quad and\quad C\quad =\quad \left[ \begin{matrix} 6 \\ -2 \end{matrix} \right]$$
$$B+C$$

if $$A = [ 8\ \ -3]$$ and $$B = [ 4\ \ -5]$$ find 
$$(i) A +B$$

Compute the following
$$\begin{bmatrix} a & b \\ -b & a \end{bmatrix}+\begin{bmatrix} a & b \\ b & a \end{bmatrix}$$

Evaluate  :
$$7\begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}$$

Evaluate  :
$$3 [ 5-2 ]$$

Given $$A\quad =\begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}\quad and\quad B\quad =\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}$$ Find
$$A^2$$

Compute the indicated products 
$$\begin{bmatrix} a & b \\ -b & a \end{bmatrix}\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$

Compute the indicated products 
$$\left[ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right] \left[ \begin{matrix} 2 & 3 & 4 \end{matrix} \right]$$

If $$A = \begin{bmatrix} 1 & 0\\ 2 & -7 \end{bmatrix}; B = \begin{bmatrix} 3 & 8\\ 7 & 5 \end{bmatrix}; C = \begin{bmatrix} -8 & -3\\ 0 & 9 \end{bmatrix},$$ then A (B + C) =

If $$\begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix},  B=\begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}$$  and  $$(A+B)^2=A^2+B^2$$ ,  find $$a$$ and $$b$$ .

If   $$A=\begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix}$$  and   $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},$$ then find the $$k$$ so that $$A^{2}=8A+kI.$$

If $$A\times \begin{bmatrix} 1 & 2 \\ 0 & 2 \end{bmatrix}=(1, 2)$$ , then $$A=$$

If $$A=\begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}$$ and $$B=\begin{bmatrix} 6 & 7 \\ 8 & 9 \end{bmatrix}$$ ,  verify that $$(AB)^{-1}=B^{-1}A^{-1}$$

Determine all $$2\times 2$$ matrices A such that $$A^{2}=I.$$

If $$A=\begin{bmatrix} x & 1 \\ 1 & 0 \end{bmatrix}$$ and $$A^{2}=I,$$ then find $$x.$$

Find the value of $$(x \times y)$$ if $$\begin{bmatrix}3 & -2\\ 3 & 0\\ 2 & 4\end{bmatrix} \begin{bmatrix}y & y\\ x & x\end{bmatrix} = \begin{bmatrix}3 & 3\\ 3y & 3y\\ 10 & 10\end{bmatrix}.$$

The elements on the main diagonal of a skew-symmetric matrix are all ...........

If $$A = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & 0 \\ 3 & 6 & 0 \end{bmatrix}$$ and  $$B = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$ and  $$C = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix}$$ , Then evaluate matrix $$AB - BC$$

If $$A = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & 0 \\ 3 & 6 & 0 \end{bmatrix}$$ and  $$B = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$ and  $$C = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix}$$ , Then evaluate $$AB - BC$$

If $$A = \begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix}1 & -2 \\ -1 & 0 \\ 2 & -1\end{bmatrix}$$ , Check $$AB=BA$$ ? 

By the method of matrix inversion, solve the system 
$$\begin{bmatrix}1&1&1 \\ 2&5&7 \\ 2&1&-1\end{bmatrix}\begin{bmatrix}x&u \\ y&v \\ z&w\end{bmatrix} = \begin{bmatrix}9&2 \\ 52&15 \\ 0&-1\end{bmatrix}$$
and find the value of $$xyz-uvw$$

Find the inverse of $$A = \begin{bmatrix}0& 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}$$ if 
$$A^{-1} = \quad \begin{bmatrix}1/2 & -1/2 & 1/2 \\ a & 3 & b \\ c & -3/2 & 1/2\end{bmatrix}$$
Find $$|abc|?$$

Let $$A =\begin{bmatrix}1 &1  &3 \\5 &2  &6 \\-2 &-1  &-3 \end{bmatrix}.$$ Find the smallest $$n\in N,$$ such that $$A^n=O$$ .

Show that, $$\displaystyle \begin{bmatrix}1 &-\tan \frac{\theta }{2} \\ \tan \frac{\theta }{2}&1 \end{bmatrix} \begin{bmatrix}1 &\tan \frac{\theta }{2} \\ -\tan \frac{\theta }{2}&1 \end{bmatrix}^{-1}=\begin{bmatrix}\cos \theta  & -\sin \theta \\ \sin \theta& \cos \theta \end{bmatrix}.$$

If $$A =\begin{bmatrix} 1&0  &2 \\0 &2  &1 \\2  &0  &3 \end{bmatrix}$$ , then show that the matrix A is a root of the polynomial $$f(x) = x^3 - 6x^2 + 7x + 2$$ .

If $$A =$$ $$\begin{bmatrix}2 &-3 \\ 4 & 1\end{bmatrix},\, B\, =\, \begin{bmatrix}2 & 3 \\ 5 & 0\end{bmatrix}\,$$ and $$\, C\, =\, \begin{bmatrix}-1 & 2 \\ 0 & 5\end{bmatrix}$$ , then find $$A (B + C)$$ .

Find $$X$$ and $$Y$$ , if
(i)  $$\displaystyle X+Y=\begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$$ and  $$\displaystyle X-Y=\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$$
(ii) $$\displaystyle 2X+3Y=\begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix}$$ and   $$\displaystyle 3X+2Y=\begin{bmatrix} 2 & -2 \\ -1 & 5 \end{bmatrix}$$

Let $$\displaystyle A=\begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix},B=\begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix},C=\begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$$
Find each of the following
(i) $$A+B$$
(ii) $$A-B$$
(iii) $$3A-C$$
(iv) $$AB$$
(v) $$BA$$

Compute the indicated products
(i)  $$\displaystyle \begin{bmatrix} a & b \\ -b & a \end{bmatrix}\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$

(ii) $$\displaystyle \left[ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right] \left[ \begin{matrix} 2 & 3 & 4 \end{matrix} \right]$$

(iii) $$\displaystyle \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{matrix} \right]$$

(iv) $$\displaystyle \left[ \begin{matrix} 2 \\ 3 \\ 4 \end{matrix}\begin{matrix} 3 \\ 4 \\ 5 \end{matrix}\begin{matrix} 4 \\ 5 \\ 6 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 0 \\ 3 \end{matrix}\begin{matrix} -3 \\ 2 \\ 0 \end{matrix}\begin{matrix} 5 \\ 4 \\ 5 \end{matrix} \right]$$

(v) $$\displaystyle \left[ \begin{matrix} 2 \\ 3 \\ -1 \end{matrix}\begin{matrix} 1 \\ 2 \\ 1 \end{matrix} \right] \left[ \begin{matrix} 1 & 0 & 1 \\ -1 & 2 & 1 \end{matrix} \right]$$

(vi) $$\displaystyle \left[ \begin{matrix} 3 & -1 & 3 \\ -1 & 0 & 2 \end{matrix} \right] \left[ \begin{matrix} 2 \\ 1 \\ 3 \end{matrix}\begin{matrix} -3 \\ 0 \\ 1 \end{matrix} \right]$$

If  $$\displaystyle A=\begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$$ , then prove  $$\displaystyle { A }^{ n }=\begin{bmatrix} 1+2n & -4n \\ n & 1-2n \end{bmatrix}$$ where n is any positive integer

Given :  $$\displaystyle 3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}$$
Find the values of $$x,y,z$$ and $$w$$ .

If $$\displaystyle A=\left[ \begin{matrix} 1 \\ 5 \\ 1 \end{matrix}\begin{matrix} 2 \\ 0 \\ -1 \end{matrix}\begin{matrix} -3 \\ 2 \\ 1 \end{matrix} \right] ,B=\left[ \begin{matrix} 3 \\ 4 \\ 2 \end{matrix}\begin{matrix} -1 \\ 2 \\ 0 \end{matrix}\begin{matrix} 2 \\ 5 \\ 3 \end{matrix} \right]$$ and  $$\displaystyle C=\left[ \begin{matrix} 4 \\ 0 \\ 1 \end{matrix}\begin{matrix} 1 \\ 3 \\ -2 \end{matrix}\begin{matrix} 2 \\ 2 \\ 3 \end{matrix} \right]$$ , then compute $$(A+B)$$ and $$(B-C)$$ . 

Also, verify that  $$\displaystyle A+\left( B-C \right) =\left( A+B \right) -C$$

For what values of $$x$$ .

$$\begin{bmatrix} 1 &2&1\end{bmatrix}\begin{bmatrix} 1&2&0\\2&0&1\\1&0&2\end{bmatrix}\begin{bmatrix}0\\2\\x \end{bmatrix} = 0$$ ?

If A= $$\displaystyle \begin{bmatrix} 2 & -3   \\ 4 & 1   \end{bmatrix}$$ B= $$\displaystyle \begin{bmatrix} 2 & 3   \\ 5 & 0   \end{bmatrix}$$ and C= $$\displaystyle \begin{bmatrix} -1 & 2   \\ 0 & 5   \end{bmatrix}$$ , then find $$A(B+C)$$ .

Show that
$$(i)$$ $$\begin{bmatrix} 5 &  -1 \\ 6 &  7 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 3 & \ 4 \end{bmatrix} \neq \begin{bmatrix} 2 &  1 \\ 3 &  4 \end{bmatrix}\begin{bmatrix} 5 & -1 \\ 6 &  7 \end{bmatrix}$$

$$(ii)$$ $$\left[ \begin{matrix} 1 \\ 0 \\ 0 \end{matrix}\quad \begin{matrix} 2 \\ 1 \\ 1 \end{matrix}\quad \begin{matrix} 3 \\ 0 \\ 0 \end{matrix} \right] \left[ \begin{matrix} -1 \\ 0 \\ 2 \end{matrix} \begin{matrix} 1 \\ -1 \\ 3 \end{matrix}\quad \begin{matrix} 0 \\ 1 \\ 4 \end{matrix} \right]  \neq  \left[ \begin{matrix} -1 \\ 0 \\ 2 \end{matrix}\quad \begin{matrix} 1 \\ -1 \\ 3 \end{matrix}\quad \begin{matrix} 0 \\ 1 \\ 4 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 0 \\ 0 \end{matrix}\quad \begin{matrix} 2 \\ 1 \\ 1 \end{matrix}\quad \begin{matrix} 3 \\ 0 \\ 0 \end{matrix} \right]$$

If  $$\displaystyle A=\left[ \begin{matrix} 1 \\ 1 \\ 1 \end{matrix}\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}\begin{matrix} 1 \\ 1 \\ 1 \end{matrix} \right]$$ , prove that  $$\displaystyle { A }^{ n }=\left[ \begin{matrix} { 3 }^{ n-1 } \\ { 3 }^{ n-1 } \\ { 3 }^{ n-1 } \end{matrix}\begin{matrix} { 3 }^{ n-1 } \\ { 3 }^{ n-1 } \\ { 3 }^{ n-1 } \end{matrix}\begin{matrix} { 3 }^{ n-1 } \\ { 3 }^{ n-1 } \\ { 3 }^{ n-1 } \end{matrix} \right] ,n\in N$$

Find the matrix $$X$$ so that  $$\displaystyle X\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} \right] =\left[ \begin{matrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{matrix} \right]$$

Find $$x$$ , if $$\displaystyle \begin{bmatrix} x&-5&-1 \end{bmatrix}\begin{bmatrix}1&0&2\\0&2&1\\2&0&3 \end{bmatrix}\begin{bmatrix}x\\4\\1\end{bmatrix}=0$$

If  $$\begin{bmatrix}9 &-1  &4 \\  -2&1  &3 \\ \end{bmatrix}= A+\begin{bmatrix}1 &2  &-1 \\  0&4  &9 \\ \end{bmatrix}$$ , then find the matrix $$A$$  

If matrix $$A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$$ and $$A^2 = KA$$ , then write the value of $$K.$$

Simplify:
$$\cos \theta \begin{bmatrix} \cos \theta & \sin \theta\\ -\sin \theta & \cos \theta \end{bmatrix} + \sin \theta \begin{bmatrix}\sin \theta & -\cos \theta\\ \cos \theta & \sin \theta\end{bmatrix}.$$

The monthly incomes of Aryan and Babban are in the ratio $$3 : 4$$ and their monthly expenditures are in the ratio $$5 : 7$$ . If each saves Rs. $$15000$$ per month, find their monthly incomes using matrix method. This problem reflects which value?

If $$\begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix} \begin{bmatrix} 1 & -3 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} -4 & 6 \\ -9 & x \end{bmatrix}$$ , write the value of $$x$$ .

Matrix $$A = \begin{bmatrix} 0& 2b & -2\\ 3 & 1 & 3\\ 3a & 3 & -1\end{bmatrix}$$ is given to be symmetric, find values of $$a$$ and $$b$$

If $$\begin{bmatrix} x - y& z\\ 2x - y & w\end{bmatrix} = \begin{bmatrix} -1& 4\\ 0 & 5\end{bmatrix}$$ , find the value of $$x + y.$$

If $$A$$ is a square matrix such that $$A^{2} = A$$ , then write the value of $$7A - (I + A)^{3}$$ , where $$I$$ is an identity matrix.

If $$A=\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}$$ , find $$A^2-5A+4I$$ and hence find a matrix $$X$$ such that $$A^2-5A+4I+X=0$$ .

Solve the following matrix equation for $$x : [x  \space  1] \begin{bmatrix} 1& 0\\ -2 & 0\end{bmatrix} = 0.$$

Three school A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs $$25$$ , Rs $$100$$ and Rs $$50$$ each. The number of articles sold are given below:

Articles	School A	School B	School C
Hand-fans	$$40$$	$$25$$	$$35$$
Mats	$$50$$	$$40$$	$$50$$
Plates	$$20$$	$$30$$	$$40$$

Find the fund collected by each school separately by selling the above articles. Also, find the total funds collected for the purpose.

To promote the making of toilets for women, an organization tried to generate awarencess through:

(i) house calls

(ii) letters

(iii) announcements.
The cost for each mode per attempt is given below:
(i) Rs $$50$$
(i) Rs $$20$$
(i) Rs $$40$$
The number of attempts made in three villages X, Y and Z are given below:

	(i)	(ii)	(iii)
X	$$400$$	$$300$$	$$100$$
Y	$$300$$	$$250$$	$$75$$
Z	$$500$$	$$400$$	$$150$$

Find the total cost incurred by the organization for the three villages separately, using matrices.
Write one value generated by the organization in the society.

Given $$\begin{bmatrix}2&1\\-3&4\end{bmatrix}X=\begin{bmatrix}7\\6\end{bmatrix}$$ . Write:
(i) the order of the matrix $$X$$
(ii) the matrix $$X$$ .

Find $$x+y$$ if $$\begin{bmatrix} x & 3x\\y & 4y\end{bmatrix}$$ $$\begin{bmatrix}2 \\ 1 \end{bmatrix}$$ $$=\begin{bmatrix}5 \\1 2\end{bmatrix}$$ .

If $$A=\begin{bmatrix} 3 & 7 \\ 2 & 4 \end{bmatrix}$$ , $$B=\begin{bmatrix} 0 & 2 \\ 5 & 3 \end{bmatrix}$$ and $$C=\begin{bmatrix} 1 & -5 \\ -4 & 6 \end{bmatrix}$$ . Find $$AB-5C$$ .

Find $$x+ y$$ if $$\begin{bmatrix}-2 & 0\\ 3 & 1\end{bmatrix} \begin{bmatrix} -1\\ 2x \end{bmatrix} + 3 \begin{bmatrix} -2\\ 1\end{bmatrix} = 2 \begin{bmatrix}y \\ 3\end{bmatrix}.$$

Given A $$=\begin{bmatrix} 2 & -6\\2 & 0\end{bmatrix}, B=\begin{bmatrix} -3 & 2\\ 4 & 0\end{bmatrix}, C=\begin{bmatrix} 4 & 0\\0 & 2\end{bmatrix}$$ . Find the matrix X such that $$A+2X=2B+C$$ .

If $$A=\begin{bmatrix}3&1\\-1&2\end{bmatrix}$$ and $$I=\begin{bmatrix}  1& 0 \\ 0& 1 \end{bmatrix}$$ , find $$A^2-5A+7I$$

Given $$A = \begin{bmatrix}2 & 0\\ -1 & 7\end{bmatrix}$$ and $$I = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$$ and $$A^{2} = 9A + mI$$ . Find $$m.$$

Given matrix $$A = \begin{bmatrix}4\sin 30^{\circ} & \cos 0^{\circ} \\ \cos 0^{\circ} & 4\sin 30^{\circ} \end{bmatrix}$$ and $$B = \begin{bmatrix}4\\5  \end{bmatrix}$$ . If $$AX = B$$ .
Find the matrix $$X$$ .

Find $$A$$ , if $$A + B = \begin{bmatrix}5 & 2\\ 0 & 9\end{bmatrix}$$ and $$A - B = \begin{bmatrix}-3 & -6\\ 4 & -1\end{bmatrix}$$

If $$A=\begin{bmatrix} \cos { \alpha  }  & \sin { \alpha  }  \\ -\sin { \alpha  }  & \cos { \alpha  }  \end{bmatrix}$$ , then find $${A}^{2}$$ .

Let $$A = \begin{bmatrix} 2& 1\\ 0 & -2\end{bmatrix}, B = \begin{bmatrix}4 & 1\\ -3 & -2\end{bmatrix}$$ and $$C = \begin{bmatrix}-3 & 2\\ -1 & 4\end{bmatrix}$$ . Find $$A^{2} + AC - 5B$$

If $$A = \begin{bmatrix}-2\\ 4\\ 6 \end{bmatrix}$$ and $$B = [1\  4\ -6]$$ then find $$AB$$

Find the matrix $$B$$ , such that
$$\begin{bmatrix} 4 & 3 \\ 3 & 2 \end{bmatrix}B=\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$$

Find the matrix for which:
$$\begin{bmatrix} 5 & 4 \\ 1 & 1 \end{bmatrix}X=\begin{bmatrix} 1 & -2 \\ 1 & 3 \end{bmatrix}$$

Solve for x and y if $$\left(\begin{matrix} {y^2}\\ {x^2} \end{matrix}\right) + 2 \left( \begin{matrix}{3y}\\{2x}\end{matrix} \right )  = 3 \left (\begin{matrix}{-3}\\{7} \end{matrix}\right )$$

If $$2A + B = \begin{bmatrix} 3& -1\\ 2 & 4\end{bmatrix}$$ and $$B = \begin{bmatrix}-1 & -5\\ 0 & 2\end{bmatrix}$$ , then find $$A$$ .

If $$(A - 2I)(A - 3I) = 0$$ , where $$A =\begin{bmatrix}4 & 2\\ -1 & x\end{bmatrix}$$ and  $$I =\begin{bmatrix}1& 0\\ 0& 1 \end{bmatrix}$$  , find the value of $$x$$ .

If $$A = \begin{bmatrix}2 & 0 & 1\\ 2& 1 & 3\\ 1 & -1 & 0\end{bmatrix}$$ , then find the value of $$A^{2} - 5A + 6I$$

Find the matrix $$X$$ , given that $$X + 2I = \begin{bmatrix}3 & -1\\ 1 & 2\end{bmatrix}$$

If $$A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ ; $$B=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}$$ and $$2X + A = B$$ , then find $$X$$ .

If $$A=\begin{bmatrix} 2 & 5 \\ -1 & 0 \end{bmatrix}$$ and $$B=\begin{bmatrix} 2 & 3 \\ -2 & 4 \end{bmatrix}$$ , find the matrix $$P$$ when $$2A + P = B$$ .

If $$A = \begin{bmatrix}a &b \\ c & d\end{bmatrix}$$ and $$I = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$$ , then show that $$A^{2} - (a + d) A = (bc - ad)I$$

Find the product of the matrices $$\begin{pmatrix} 3& -2\\ 5 & 1\end{pmatrix} \begin{pmatrix}4 & 1\\ 2 & 7\end{pmatrix}$$

If $$A=\begin{bmatrix} 2 & 4 \\ -1 & k \end{bmatrix}$$ and $${ A }^{ 2 }=O$$ (null matrix), then find the value of $$k$$ .

If $$A = \begin{bmatrix}1 &4 \\ 2 &1 \end{bmatrix}, B = \begin{bmatrix}-3 & 2\\ 4 & 0\end{bmatrix}, C = \begin{bmatrix}1 & 0\\ 0 & 2\end{bmatrix}$$ ; find $$A^{2} + BC$$

Let $$A = \bigl(\begin{smallmatrix}3 & 2\\ 5  & 1\end{smallmatrix}\bigr)$$ and $$B = \bigl(\begin{smallmatrix}8 & -1\\ 4  & 3\end{smallmatrix}\bigr)$$ . Find the matrix C if $$C = 2A + B$$ .

Matrix A shows the weight of four boys and four girls in kg at the beginning of a diet programme to lose weight. Matrix B shows the corresponding weights after the diet programme.
$$A = \begin{bmatrix}35 & 40 & 28 & 45\\ 42 & 38 & 41 & 30\end{bmatrix}\underset{Girls}{Boys}, B =  \begin{bmatrix}32 & 35 & 27 & 41\\ 40 & 30 & 34 & 27\end{bmatrix} \underset{Girls}{Boys}$$
Find the weight loss of the Boys and Girls.

If $$A = \bigl(\begin{smallmatrix}-1 & 2 & 1 \\ 1 & 2 & 3\end{smallmatrix}\bigr), B = \bigl(\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\bigr)$$ and $$C = \bigl(\begin{smallmatrix} 2 & 1\end{smallmatrix}\bigr)$$ verify (AB)C = A(BC).

Find the product of the matrices, if exists.

$$\bigl(\begin{smallmatrix} 3& -2 \\ 5 & 1\end{smallmatrix}\bigr) \bigl(\begin{smallmatrix}4  & 1\\ 2 & 7 \end{smallmatrix}\bigr)$$

Find the product of the matrices, if exists.

$$\left(\begin{matrix} 6\\ -3\end{matrix}\right) \left(\begin{matrix} 2 & 7\end{matrix}\right)$$

The amount of fat, carbohydrate and protein in grams present in each food item respectively are as follows

	Item 1	Item 2	Item 3	Item 4
Fat	5	0	1	10
Carbohydrate	0	15	6	9
Protein	7	1	2	8

Use the information to write $$3 \times 4$$ and $$4 \times 3$$ matrices.

Find the product of the matrices, if exists.

$$\bigl(\begin{smallmatrix}2 & 9 & -3\\ 4 & -1 & 0\end{smallmatrix}\bigr) \bigl(\begin{smallmatrix} 4& 2 \\ -6  & 7 \\ -2 & 1\end{smallmatrix}\bigr)$$

If $$A = \begin{bmatrix}5 & 6 & -2  & 3\\ 1 & 0 & 4 & 2\end{bmatrix}$$ and $$B = \begin{bmatrix}3 & -1 & 4 & 7\\ 2 & 8 & 2 & 3\end{bmatrix}$$ , then find A + B.

If $$A = \bigl(\begin{smallmatrix} 8& -7 \\ -2  &4 \\ 0 & 3 \end{smallmatrix}\bigr)$$ and $$B = \bigl(\begin{smallmatrix} 9& -3 & 2\\ 6 & -1 & -5\end{smallmatrix}\bigr)$$ , then find AB and BA if they exists.

If $$A=\begin{bmatrix}1&2\\2&1\end{bmatrix}.B = \begin{bmatrix}2&0\\1&3\end{bmatrix}$$ and $$C=\begin{bmatrix}1&1\\2&3\end{bmatrix}$$ calculate $$AC, BC$$ and $$(A+B)C$$ . Also verify that $$(A+B)C=AC+BC.$$

If $$M=\begin{bmatrix}7 & 5\\2 & 3\end{bmatrix}$$ , then verify the equation $$M^2-10M+11 I_2=0$$ .

For matrices $$A=\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$$ and $$B=\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$ , write $$A+B$$ and $$A-B$$ .

(i)   $$2i-j$$ is the $$(i,j)$$ th element of a $$2\times 2$$ matrix $$A$$ .
(ii)  Find the matrix $$B$$ such that $$3A-2B=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
(iii)   Prove that $${ A }^{ 2 }+2I=3A$$

Given matrix $$B=\begin{bmatrix} 1 & 1\\ 8 & 3\end{bmatrix}$$ . Find the matrix X if, $$X=B^2-4B$$ . Hence solve for a and b given $$X\begin{bmatrix} a \\ b\end{bmatrix}=\begin{bmatrix} 5 \\ 50\end{bmatrix}$$ .

If the matrix $$\begin{pmatrix}6 & -x^2\\ 2x-15 & 10\end{pmatrix}$$ is symmetric, find the value of $$x$$ .

If $$A=\begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta\end{bmatrix}$$ . Find the value $$A^2$$ .

Find the inverse of $$A = \begin{bmatrix} \cos \alpha& -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0 & 0 & 1\end{bmatrix}$$ using elementary transformation.

If $$A =\begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$$ then find $$A^{-1}$$

If $$A = \begin{bmatrix} 0& 6 & 7\\ -6 & 0 &8 \\ 7 & -8 & 0\end{bmatrix}, B = \begin{bmatrix}0 & 1 & 1\\ 1 & 0 & 2\\ 1 & 2 & 0\end{bmatrix}, C = \begin{bmatrix}2 \\ -2 \\  3\end{bmatrix}$$ , calculate $$AC, BC$$ and $$(A + B)C$$ .
Also, verify that $$(A + B)C = AC + BC$$ .

If $$A=\begin{bmatrix}  \cos \alpha & -\sin \alpha\\ \sin \alpha  & \cos \alpha \end{bmatrix}$$ , $$B=\begin{bmatrix}  \cos 2\beta & \sin 2\beta\\ \sin 2\beta  & -\cos 2\beta \end{bmatrix}$$ , where $$0<\beta<\large{\frac{\pi}{2}}$$ , then prove that $$BAB=A^{-1}$$ . .

Is it possible to define the matrix A + B when 
A has 3 rows and B has 2 rows

If $$A=\begin{bmatrix}x&y&z\end{bmatrix},$$ $$B=\begin{bmatrix}a&h&g\\h&b&f\\g&f&c\end{bmatrix}, C=\begin{bmatrix}\alpha & \beta & \gamma \end{bmatrix}^{T}$$ then $$ABC$$ is

Using elementary row transformation, find the inverse of the matrix $$A=\begin{bmatrix}1&2&3\\2&5&7\\-2&-4&-5\end{bmatrix}$$ .

Is it possible to define the matrix A + B when 
A has 3 rows and B has 2 columns 

If $$A = \begin{vmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{vmatrix}$$ , then prove that $$A^2 - 4A - 5 I_3 = 0$$ and hence obtain $$A^{-1}$$ .

If $$A=\begin{bmatrix}1&1&1\\0&1&1\\ 0&0&1\end{bmatrix},$$ show that $$A^n=\begin{bmatrix}1&n&\dfrac{n(n+1)}{2}\\0&1&n\\0&0&1\end{bmatrix}\forall n \in \mathbb{N}$$ . 

Is it possible to define the matrix A + B when 
A has 2 columns and B has 4 columns

If $$\omega = e^{2i\pi /3}$$ . Find the inverse of the matrix.
A = $$\begin{bmatrix}               1 & 1 & 1 \\[0.3em]               1 & \omega & \omega^2 \\[0.3em]               1 & \omega^2 & \omega               \end{bmatrix}$$

Compute $$A^2 \, - \, 4A \, - \, 5$$  Where A = $$\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 1 \\ 2 & 2 & 1\end{bmatrix}$$
and I is a unit matrix

If A + B = $$\begin{bmatrix}1 & 0 & 2\\ 2 & 2 & 2\\ 1 & 1 & 1 \end{bmatrix}$$ and A - B  =   $$\begin{bmatrix}1 & 4 & 4\\ 4 & 2 & 0\\ -1 & -1 & 2 \end{bmatrix}$$
then prove that 
A = $$\begin{bmatrix}1 & 2 & 3\\ 3 & 2 & 1\\ 0 & 0 & 2 \end{bmatrix}$$ and   B  =   $$\begin{bmatrix}0 & -2 & -1\\ -1 & 0 & 1\\ 1 & 1 & 0 \end{bmatrix}$$

If $$A_\alpha \, = \, \begin{bmatrix}cos \, \, n\alpha & sin \, n\alpha \\ -sin \, n\alpha  & cos \, \,  n\alpha \end{bmatrix}$$ , then prove the following 
$$(A_\alpha)^n \, = \, \begin{bmatrix}cos \, \, n\alpha & sin \, n\alpha \\ -sin \, n\alpha  & cos \, \,  n\alpha \end{bmatrix}$$

Find the inverse of the following matrices:
(a) $$\begin{bmatrix}               1 & 4 & 0 \\[0.3em]               -1 & 2 & 2 \\[0.3em]               0 & 0 & 2               \end{bmatrix}$$
(b) $$\begin{bmatrix}               cos\theta & -sin\theta & 0 \\[0.3em]               sin\theta & cos\theta & 0 \\[0.3em]               0 & 0 & 1               \end{bmatrix}$$
(c)  $$\begin{bmatrix}               2 & 2 & 2 \\[0.3em]               2 & 5 & 5 \\[0.3em]               2 & 5 & 11               \end{bmatrix}$$
(d) $$\begin{bmatrix}               1 & 2 & 3 \\[0.3em]               2 & 4 & 5 \\[0.3em]               3 & 5 & 6               \end{bmatrix}$$
(e)  $$\begin{bmatrix}               1 & 4 & 0 \\[0.3em]               -1 & 2 & 2 \\[0.3em]               0 & 0 & 2               \end{bmatrix}$$

If A  is a diagonal  matrix  diag $$(d_1 , d_2 \, , ...... d_n )$$   then $$A^n  , n    \epsilon   R\  is\ diag  (d^n_1 \, , d^n_2 \, .....d^n_n )$$

$$\left( {\matrix{    1 & 3 & { - 2}  \cr    { - 3} & 0 & 1  \cr    2 & 1 & 0  \cr  } } \right)$$
Find Inverse Elementary transformation

Find the inverse of matrix
$$\begin{bmatrix}               14 & 3 & -2 \\[0.3em]               6 & 8 & -1 \\[0.3em]               0 & 2 & -7               \end{bmatrix}.$$

If $$A=\begin{bmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1 \end{bmatrix},B=\begin{bmatrix} 3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3 \end{bmatrix},C=\begin{bmatrix} 4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3 \end{bmatrix}$$ , Then Compute $$(A+B)$$ and $$(B-C)$$ . Also, verify that $$A+(B-C)=(A+B)-C$$

Find matrix $$X$$ so that $$X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6\end {bmatrix}$$ .

If $$A=\left[ \begin{matrix} -1 & 2 & 0 \\ -1 & 1 & 1 \\ 0 & 1 & 0 \end{matrix} \right]$$ , Show that $$\\ { A }^{ 2 }={ A }^{ -1 }$$

Using elementary row transformations, find the inverse of the matrix

$$\begin{bmatrix}1 & 2 & 3\\  2& 5 &7 \\ -2 &-4  & 5\end{bmatrix}$$

Given $$F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ If $$x \in R$$ Then prove $$y$$ , $$F(x + y) = F(x) F(y)$$

If $$f\left( \alpha  \right) =\begin{bmatrix} \cos { \alpha  }  & -\sin { \alpha  }  & 0 \\ \sin { \alpha  }  & \cos { \alpha  }  & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ ,
Find
(i) $$f\left( -\alpha  \right)$$
(ii) $$f\left( -\alpha  \right) +f\left( \alpha  \right)$$ .

If $$A=\begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}$$ and $$\begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 5 & 1 \end{bmatrix}$$ and $$BA=(b_0)$$ , then find $$b_{21}+b_{32}$$

Let $$A=\begin{bmatrix}0&1\\1&0\end{bmatrix}$$ then $$A^4=$$ .

If $$\omega$$ is complex (non real) cube root of $$1$$ then show that $$\begin{vmatrix} 1 & \omega  & { \omega  }^{ 2 } \\ \omega  & { \omega  }^{ 2 } & 1 \\ { \omega  }^{ 2 } & 1 & \omega  \end{vmatrix}=0$$ .

If $$A=\begin{bmatrix} 0 & -\tan { \dfrac { \alpha  }{ 2 }  }  \\ \tan { \dfrac { \alpha  }{ 2 }  }  & 0 \end{bmatrix}$$ and $$I$$ is the identity matrix of order $$2$$ , show that $$I+A=\left( I-A \right) \begin{bmatrix} \cos { \alpha  }  & -\sin { \alpha  }  \\ \sin { \alpha  }  & \cos { \alpha  }  \end{bmatrix}$$

$$AB=0$$ where
$$A=\begin{bmatrix} \cos ^{ 2 }{ \theta  }  & \cos { \theta  } \sin { \theta  }  \\ \cos { \theta  } \sin { \theta  }  & \sin ^{ 2 }{ \theta  }  \end{bmatrix}$$
$$B=\begin{bmatrix} \cos ^{ 2 }{ \phi  }  & \cos { \phi  } \sin { \phi  }  \\ \cos { \phi  } \sin { \phi  }  & \sin ^{ 2 }{ \phi  }  \end{bmatrix}$$
then find $$\left| \theta -\phi  \right| =?$$

Given $$A=\left[ \begin{matrix} 1 & 1 & 1 \\ 2 & 4 & 1 \\ 2 & 3 & 1 \end{matrix} \right] ,B=\begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}$$ . Find $$P$$ such that $$BPA=\left[ \begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right]$$ . 

Find matrices $$A$$ and $$B$$ , where
(i) $$2A-B=\begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$$ and $$A+3B=\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$
(ii) $$3A-B=\left[ \begin{matrix} -1 & 2 & 1 \\ 1 & 0 & 5 \end{matrix} \right]$$ an $$A+5B=\left[ \begin{matrix} 0 & 0 & 1 \\ -1 & 0 & 0 \end{matrix} \right]$$

Given $$M\times \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}=\left[ \begin{matrix} 1 & 2 \end{matrix} \right]$$  
Find:
(i)The order of matrix $$M$$ ?
(ii)The matrix $$M$$ . ?

Using elementary tranormtion, find the inverse of $$\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix}$$

If $$3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}$$ , find the values of $$x,y,z$$ & $$w$$ .

If $$M = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$$ and $$M^2 - \lambda M - I = O,$$ then $$\lambda =$$ ____.

If $${A=[\begin{matrix} 2 & 4 \\ -1 & k \end{matrix}]}$$ and $${A}^{2}=0$$ , find $$k$$

Given that $$\begin{bmatrix}1&x\end{bmatrix} \begin{bmatrix}2& -1\\1&2\end{bmatrix} \begin{bmatrix}1\\3\end{bmatrix} =O.$$ Find $$7x.$$

Find a $$2 \times 2$$ matrix A such that $$A \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} = 6I_2$$

If $$\theta-\phi=\pi/2$$ , then show that $$\begin{bmatrix} \cos ^{ 2 }{ \theta  }  & \cos { \theta  } \sin { \theta  }  \\ \cos { \theta  } \sin { \theta  }  & \sin ^{ 2 }{ \theta  }  \end{bmatrix}\begin{bmatrix} \cos ^{ 2 }{ \varphi  }  & \cos { \varphi  } \sin { \varphi  }  \\ \cos { \varphi  } \sin { \varphi  }  & \sin ^{ 2 }{ \varphi  }  \end{bmatrix}=0$$

If $$\theta - \phi = \pi/2$$ then show that $$\begin{bmatrix} \cos^{2} \theta& \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin^{2} \theta \end{bmatrix}\begin{bmatrix} \cos^{2}\phi& \cos \phi \sin \phi \\ \cos \phi \sin \phi & \sin \phi\end{bmatrix} = 0$$

If $$P=\begin{bmatrix} 1 & 2 & -3 \\ 3 & -1 & 2\\ -2 & 1 & 3\end{bmatrix}$$ , $$Q=\begin{bmatrix} 2 & 3 & 1\\ 3 & 1 & 2\\ 1 & 2 & 3\end{bmatrix}$$ , then find matrix R such that $$P+Q+R$$ is a zero matrix.

If $$A=\begin{bmatrix} 2 & 0 & 1\\ 2 & 1 & 3\\ 1 & -1 & 0\end{bmatrix}$$ find $$A^2-5A+4I$$ and hence find a matrix X such that $$A^2-5A+4I+X=0$$ .

If the system of equations $$3x-2y+z=0,\lambda x-14y+15z=0,x+2y-3z=0$$ have non zero solution, then find $$\lambda$$

If $$A=$$ diagonal $$[1,-2,5],B=$$ diagonal $$[3,0,-4]$$ and $$C=$$ diagonal $$[-2,7,0]$$ , then find $$A+2B-3C$$ .

If $$A = \left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&{ - 2}\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}2&a\\{ - 1}&b\end{array}} \right]$$ and if $${\left( {A + B} \right)^2} = {A^2} + {B^2}$$ , find the values of $$a$$ and $$b$$ .

Show that the matrix $$A=\begin{bmatrix} 5 & 3 \\ 12 & 7 \end{bmatrix}$$ satisfies the equation $$A^{2}-12A-I=0$$

Find the inverse of $$\begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & -1 \\ 3 & -5 & 0 \end{bmatrix}$$ by using elementary row transformations.

[2x  3] $$\begin{bmatrix} 1 & 2\\ -3 & 0\end{bmatrix}\begin{bmatrix} x\\ 3\end{bmatrix}=0$$ then find x.

If A and B are two matrices of same order $$3 \times 3$$ where A= $$\begin{bmatrix}1 & 2 &3 \\ 2 & 3 &4 \\ 5 & 6 &7 \end{bmatrix}$$ and B=   $$\begin{bmatrix}3 & 2 &5 \\ 2 & 3 &8 \\ 7 & 2 &9 \end{bmatrix}$$ AB=?

Find $$x$$ and $$y$$ , when $$x + y = \left[ {\begin{array}{*{20}{c}}7&0\\2&5\end{array}} \right]$$ and $$x - y = \left[ {\begin{array}{*{20}{c}}3&0\\0&3\end{array}} \right]$$ .

Let $$A+B =I$$ , where A= $$\left[ \begin{array}{cc} 0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\\ \end {array} \right]$$ Find $$B$$

Find the vector equation of the plane passing through points $$4i-3j-k,3i+7j-10k$$ and $$2i+5j-7k$$ and show that the point $$i+2j-3k$$ lies in the plane.

Let $$A$$ be a $$3\times3$$ matrix given by $$A=[a_{ij}]$$ .
If for every column vector $$X$$ ,  $$X^T AX=0$$ and $$a_{23}=-1008$$ ,then the sum of the digits of $$a_{32}$$ is

Let $$A=\begin{bmatrix}3 & 7\\ 2 & 5\end{bmatrix}$$ & $$B = \begin{bmatrix}6 & 8\\ 7 & 9\end{bmatrix}$$ check whether $$AB=BA$$ or not.

If $$A = \begin{bmatrix}1 & -1 & 0\\ 2 & 3 & 4\\ 0 & 1 & 2\end{bmatrix}$$ and $$B = \begin{bmatrix}2 & 2 & -4\\ -4 & 2 & -4\\ 2 & -1 & 5\end{bmatrix}$$ are two square matrices, find $$AB$$ .

If $$A=\begin{bmatrix} 1 & 2\\ 3 & -4 \\ 5 & 6\end{bmatrix}$$ and $$B=\begin{bmatrix} 4 & 5 & 6\\ 7 & -8 & 2\end{bmatrix}$$ will $$AB$$ be equal to $$BA$$ . Also find $$AB$$ & $$BA$$ .

If $$A=\begin{bmatrix} 3 & -4\\ 1 & -1\end{bmatrix}$$ , then show that $$A^3=\begin{bmatrix} 7 & -12 \\ 3 & -5\end{bmatrix}$$ .

Find the matrix $$A$$ for which $$\begin{bmatrix} 2 & 1\\ 3 & 2\end{bmatrix} A\begin{bmatrix} -3 & 2\\ 5 & -3\end{bmatrix}=\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$$ .

If A = $$\left[ \begin{matrix} 1 & 2  \\ 2 & 3  \end{matrix} \right]$$ then find $${A}^{2}$$  

If $$A=\begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix}$$ , Find $$A^-1$$ . Using $$A^-1$$ , solve the following system or liner equation.
$$x+2y+z=4 ;\, -z+y+z=0;\, x-3y+z=2.$$

Find the inverse of $$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}$$ by adjoint method.

If $$A=\begin{bmatrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{bmatrix}$$ then find $$(A^{T})^{-1}$$  

If $$A = \left[ {\begin{array}{*{20}{c}}i&0\\0&i\end{array}} \right]$$  , find $${A^2}$$ .

Find $$x$$ and $$y$$ if $$\left[ \begin{array}{l}\,\,\,3\,\,\,\,\,\,\,2\\ - 1\,\,\,\,\,\,\,4\end{array} \right]\left[ \begin{array}{l}2x\\\,\,1\end{array} \right] + 2\left[ \begin{array}{l} - 4\\\,\,\,5\end{array} \right] = 4\left[ \begin{array}{l}2\\y\end{array} \right]$$

For a real number $$\alpha$$ , if the system $$\begin{bmatrix} 1 & \alpha & \alpha^2\\ \alpha & 1 & \alpha\\ \alpha^2 & \alpha & 1\end{bmatrix}\begin{bmatrix} x \\ y \\ z\end{bmatrix}=\begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix}$$ of linear equations, has infinitely many solutions, then $$1+\alpha +\alpha^2=?$$

Find inverse of the matrix $$\begin{bmatrix} 1 & -1 & 1 \\ 0 & 1 & 1 \\ 3 & 2 & -4 \end{bmatrix}$$ by elementary transformation.

If $$A = \begin{bmatrix} 1& -2 & 3\\ -4 & 2 & 5\end{bmatrix}$$ and $$B = \begin{bmatrix} 2& 3\\ 4 & 5\\2 & 1 \end{bmatrix}$$ , then find $$AB$$

To find the inverse of the matrix $$A=[1 2 1; 0 1 1; 3 1 1]$$ by elementary transformation method.

If $$A = \begin{bmatrix}0 & -\tan \dfrac {\alpha}{2}\\ \tan \dfrac {\alpha}{2} & 0\end{bmatrix}$$ and $$I$$ the identity matrix of order $$2$$ , show that $$I + A = (I - A) \begin{bmatrix}\cos \alpha & -\sin \alpha\\ \sin \alpha & \cos \alpha\end{bmatrix}$$ .

If $$x \begin{bmatrix} 2 \\ 3 \end{bmatrix}+y\begin{bmatrix} -1 \\ 1 \end{bmatrix}=\begin{bmatrix} 10 \\ 5 \end{bmatrix}$$ , find the value of $$x$$ and $$y$$ .

If $$A=\left| \begin{matrix} 0 & -\tan { \frac { \alpha  }{ 2 }  }  \\ \tan { \frac { \alpha  }{ 2 }  }  & 0 \end{matrix} \right|$$ and $$I$$ is the identity matrix of order $$2$$ , 

show that $$I+A=(I-A)\left[ \begin{matrix} \cos { \alpha  }  & -\sin { \alpha  }  \\ \sin { \alpha  }  & \cos { \alpha  }  \end{matrix} \right]$$ .

If $$2A + B = \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\\begin{array}{l}0\\1\end{array}&\begin{array}{l}1\\ - 2\end{array}\end{array}} \right]A - 2B = \left[ {\begin{array}{*{20}{c}}0&1\\\begin{array}{l} - 2\\1\end{array}&\begin{array}{l}0\\ - 1\end{array}\end{array}} \right]$$ find $$A$$ and $$B$$

Find the values of $$x,\ y,\ z$$ if the matrix $$A= \left[ \begin{matrix} 0 \\ x \\ x \end{matrix}\begin{matrix} 2y \\ y \\ -y \end{matrix}\begin{matrix} z \\ -z \\ z \end{matrix} \right]$$ satisfy the equation $$A'A=I_{3}.$$

If $$A=\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right]$$ , find $$A^{-1}$$ by elementary transformation.

Using elementary tansormations, find the inverse of each of the matrices, if it exists in 
$$\begin{bmatrix}2&3\\5&7\end{bmatrix}$$

Find the inverse of matrices by elementary raw transformations
$$\left| \begin{matrix} \cos { \theta  }  & -\sin { \theta  }  & 0 \\ \sin { \theta  }  & \cos { \theta  }  & 0 \\ 0 & 0 & 1 \end{matrix} \right|$$

Using elementary tansormations, find the inverse of each of the matrices, if it exists in 
$$\begin{bmatrix}1&3\\2&7 \end{bmatrix}$$

Using elementary tansormations, find the inverse of each of the matrices, if it exists in 
$$\begin{bmatrix}2&5\\1&3 \end{bmatrix}$$

Using elementary tansormations, find the inverse of each of the matrices, if it exists in 
$$\begin{bmatrix}2&1\\1&1 \end{bmatrix}$$

Using elementary tansormations, find the inverse of each of the matrices, if it exists in 
$$\begin{bmatrix}2&1\\7&4 \end{bmatrix}$$

Using elementary tansormations, find the inverse of each of the matrices, if it exists in 
$$\begin{bmatrix}1&-1\\2&3\end{bmatrix}$$

If $$A =  \begin{pmatrix} 1 & 0 & 2  \\  0 & 2 & 1 \\ 2 & 0 & 3 \end{pmatrix}$$ and $$A^3 - 6A^2 + 7 A + kI^3 = 0$$ , find k.

Find the inverse of the following martrices by using transformation method.
$$\left[ \begin{matrix} 2 & -3 \\ -1 & 2 \end{matrix} \right]$$

Find the inverse of the following matrices by the adjoining method
$$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$$

If $$A=\left| \begin{matrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{matrix} \right|$$ then find $$A^{-1}$$ and hence prove that $$A^{2}-4A-5I=0$$

Find the inverse of the following matrices by the adjoining method
$$\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}$$

Find the inverse of the following martrices by using transformation method.
$$\left[ \begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} \right]$$

Find the inverse of the following martrices by using transformation method.
$$\left[ \begin{matrix} 1 & 2 \\ 2 & -1 \end{matrix} \right]$$

Find the inverse of the following martrices by using transformation method.
  $$\begin{matrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{matrix}$$  

If $$A=diag[2,-3,-5],\ B=diag[4,-6,-3]$$ and $$C=diag[-3,4,1]$$ then find
$$2A+B-5C$$

A and B are matrices

Show by an example that for $$A \neq O$$ and $$B\neq O, AB =O$$ .

Find the inverse of the following matrices by the adjoining method
$$\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}$$

Find the matrices $$A$$ and $$B$$ such that $$A+B=\left[ \begin{matrix} 5 & 4 \\ 7 & 3 \end{matrix} \right]$$ and $$A-B=\left[ \begin{matrix} 11 & 2 \\ -1 & 7 \end{matrix} \right]$$

If $$A=\left[ \begin{matrix} 6 & 2 \\ 5 & -4 \end{matrix} \right]$$ and $$B=\left[ \begin{matrix} 1 & 2 \\ -5 & 1 \end{matrix} \right]$$ , find a matrix $$X$$ such that $$2A+3B-5X=0$$

If $$A=\left[ \begin{matrix} 3 & 1 \\ -1 & 2 \end{matrix} \right]$$ and $$I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right]$$ ,then show that $$A^{2}-5A+7I_{2}=0$$  

In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories ,type I,II and III are70 and 65 in factory A and 85,65 and 72 in factory B. For girls the number of units of I,II and III are 80,75.90 in factory A and 50,55,80 in factory B. Express this information in terms of matrices and using matrix algebra answer the following questions.

1.Total units of type $$I$$ produced for boys?

2.Total production of each type for boys?

3.Total production of each type for girls?

Find the inverse of matrices by
elementary row transformation.
$$A=\left| \begin{matrix} 1 & 2 & 1 \\ -1 & 0 & 2 \\ 2 & 1 & -3 \end{matrix} \right|$$
$$B=\left| \begin{matrix} cos\theta  & -sin\theta  & 0 \\ sin\theta  & cos\theta  & 0 \\ 0 & 0 & 1 \end{matrix} \right|$$
$$A^{-1}=\left( \begin{matrix} cos\theta  & sin\theta  & 0 \\ -sin\theta  & cos\theta  & 0 \\ 0 & 0 & 1 \end{matrix} \right)$$

If $$A=\begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}$$ , $$B=\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}$$ , find $$AB$$ .

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs.The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs.70.Find the cost of each item per kg by matrix method.

Evaluate:
$$\left[ \begin{matrix} 3 & 4 & 1 \end{matrix} \right] \left[ \begin{matrix} 3 \\ -1 \\ 3 \end{matrix} \right] = [?]$$  

If $$A=\left[ \begin{matrix} 2 & 2 \\ -3 & 1 \\ 4 & 0 \end{matrix} \right] ,B=\left[ \begin{matrix} 6 & 2 \\ 1 & 3 \\ 0 & 4 \end{matrix} \right]$$ , find matrix $$C$$ such that $$A+B+C=0$$ , where $$O$$ is the zero matrix.

Given $$A=\left[ \begin{matrix} p & 0 \\ 0 & 2 \end{matrix} \right]$$ and $$B=\left[ \begin{matrix} 0 & -q \\ 1 & 0 \end{matrix} \right]$$ , and $$C=\left[ \begin{matrix} 2 & -2 \\ 2 & 2 \end{matrix} \right]$$ and $$BA=C^{2}$$ , find the values of $$p$$ and $$q$$

If $$A=\begin{bmatrix} 0 & -\tan { \dfrac { \alpha  }{ 2 }  }  \\ \tan { \dfrac { \alpha  }{ 2 }  }  & 0 \end{bmatrix}$$ and $$I$$ is the unit matrix, show that $$I+A=(I-A)\begin{bmatrix} \cos { \alpha  }  & \sin { \alpha  }  \\ \sin { \alpha  }  & \cos { \alpha  }  \end{bmatrix}$$ .

If A  $$\left[ \begin{matrix} 3 & 2 & -1 \\ 2 & -2 & 0 \\ 1 & 3 & 1 \end{matrix} \right]$$ , B  $$\left[ \begin{matrix} -3 & -1 & 0 \\ 2 & 1 & 3 \\ 4 & -1 & 2 \end{matrix} \right]$$ and $$X = A+B$$ then find $$X$$

$$A = \left| {\begin{array}{*{20}{c}}1&1\\2&2\\3&{ - 1}\end{array}} \right|$$ and $$B = \left| {\begin{array}{*{20}{c}}2&1\\1&2\\{ - 1}&0\end{array}} \right|$$ , then find matrix $$C$$ if $$A+2B+C=0$$ .

If $$A = \left[ \begin{gathered}  4\,\,5 \hfill \\  2\,\,1 \hfill \\ \end{gathered}  \right]$$ , then show that $${A^{ - 1}} = \frac{1}{6}\left( {A - 5I} \right)$$ .

If A= $$\left[ \begin{matrix} 3 & 4 & -2 \\ 2 & 1 & 0 \end{matrix} \right]$$ B = $$\left[ \begin{matrix} 2 & -1 \\ 3 & 4 \\ 0 & 2 \end{matrix} \right]$$ find AB

If A= $$\left[ \begin{matrix} 2 \\ -5 \end{matrix}\begin{matrix} -4 \\ 3 \end{matrix} \right]$$ then find  $${\text{ - A + }}{{\text{A}}^{{\text{1}}\;}}\;{\text{and}}\;{\text{ - A}}{{\text{A}}^{\text{1}}}$$

If $$A= \left[ \begin{matrix} \alpha  & \beta  \\ \gamma  & -\alpha  \end{matrix} \right]$$ and $$A^{2}=I$$ . Find the relation given by $$a^{2}=I$$ .

If $$A=\begin{bmatrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{bmatrix}$$ , find $$(A')^{-1}$$ .

If $$A=\left[ \begin{matrix} 4 & 1 \\ -1 & 2 \end{matrix} \right]$$ show that $$6A-A^{2}=9I$$

If $$A=\begin{pmatrix} p & q \\ 0 & 1 \end{pmatrix}$$ , then show that 
$$\displaystyle{ A }^{ 8 }=\begin{pmatrix} { p }^{ 8 } & \quad q\left( \dfrac { { p }^{ 8 }-1 }{ p-1 }  \right)  \\ 0 & 1 \end{pmatrix}$$

Find the inverse of the following matrix using transformation method.
$$\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$$

Solve the following system of equations by matrix methods: $$x + y + z = 6; y + 3z = 11$$ and $$x - 2y + z = 0.$$

If $$A = \left[ \begin{array} { l l } { 1 } & { - 1 } \\ { 2 } & { - 1 } \end{array} \right] , B = \left[ \begin{array} { c c } { a } & { 1 } \\ { b } & { - 1 } \end{array} \right]$$ and $$( A + B ) ^ { 2 } = A ^ { 2 } + B ^ { 2 }$$ ,find $$a$$ and $$b$$ .

Find the inverse of the following matrix using transformation method.
$$\begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$$

Find $$x,y$$ if $$\begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}\begin{bmatrix} -1 \\ 2x \end{bmatrix}+3\begin{bmatrix} -2 \\ 1 \end{bmatrix}=2\begin{bmatrix} y \\ 3 \end{bmatrix}$$

Find the inverse of $$A=\begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$$ by column transformation.

Find the $${{\text{A}}^{{\text{ - 1}}}}$$ of following matrix of by elementary transformation.

(a)  $$\left[ \begin{matrix} 1 \\ 0 \\ 1 \end{matrix}\begin{matrix} 0 \\ 2 \\ 2 \end{matrix}\begin{matrix} 1 \\ 3 \\ 1 \end{matrix} \right]$$ (b)  $$\left[ \begin{matrix} 1 \\ 0 \\ -1 \end{matrix}\begin{matrix} 2 \\ -2 \\ 3 \end{matrix}\begin{matrix} -2 \\ 1 \\ 0 \end{matrix} \right]$$

Given $$3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}$$ , find the values of $$x,y,z$$ and $$w$$ .

If $$A = \begin{bmatrix}1 & 0 & 0\\ 1 & 0 & 1\\ 0 & 1 &0\end{bmatrix}$$ then for $$n \ge 4, A^n =$$

Find inverse ?
$$\left[ \begin{array} { c c c } { 2 } & { - 1 } & { 4 } \\ { 4 } & { 0 } & { 2 } \\ { 3 } & { - 2 } & { 7 } \end{array} \right]$$

Find multiplication of following matrix:
$$\begin{bmatrix} 6 & 4 \\ 3 & -1 \end{bmatrix}\begin{bmatrix} -1 \\ 3 \end{bmatrix}$$

If $$A=\begin{bmatrix} 2 & 5 \\ 4 & 4 \end{bmatrix},\quad B=\begin{bmatrix} 0 & 5 \\ 1 & 6 \end{bmatrix}$$ , find $$5A'+3B'$$

Let $$A = \left[ \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 0 } \\ { 1 } & { 1 } & { 1 } \end{array} \right]$$ and $$B = A ^ { 10 }$$
Then the sum of elements of the first column of $$B$$ is

Find the matrix $$A$$ satisfying the matrix equation $$\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}A\begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

show that $$\left| {\matrix{   { - bc} & {ca + ab} & {ca + ab}  \cr    {ab + bc} & { - ca} & {ab + bc}  \cr    {bc + ca} & {bc + ca} & { - ab}  \cr   } } \right| = {(\sum {ab} )^3}$$

If $$A=\begin{pmatrix} p & q \\ 0 & 1 \end{pmatrix}$$ , then show that $$A^{8}=\begin{pmatrix} { p }^{ 8 } & q\left( \dfrac { { p }^{ 8 }-1 }{ p-1 }  \right)  \\ 0 & 1 \end{pmatrix}$$

If $$f(x) = \left| {\matrix{   0 & {{x^2} - \sin x} & {\cos x - 2}  \cr    {\sin x - {x^2}} & 0 & {1 - 2x}  \cr    {2 - \cos x} & {2x - 1} & 0  \cr   } } \right|,$$ then  $$\int {f(x)dx}$$ is equal to 

Given, $$A=\begin{bmatrix} 1 & 3 & 5 \\ -1  & 0 & 2\\0 & 4 & -3 \end{bmatrix}$$ and $$C=\begin{bmatrix} 4 & 1 & -2  \\ 3  & 2 & 1 \\2 & -1 & 7 \end{bmatrix}$$ Find $$A+C$$

Using elementary transformation, find the inverse of the matrix $$\left[ \begin{matrix} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{matrix} \right]$$ .

To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap books and pastel sheets made by them using recycled paper at the rate of $$\square 20$$ , $$\square 15$$ , and $$\square 10$$ per unit respectively. School A sold $$25$$ paper bags, $$10$$ scrap books and $$30$$ pastel-sheets. School B sold $$20$$ paper bags, $$15$$ scrap books and $$30$$ pastel-sheets while School C sold $$25$$ paper bags, $$18$$ scrap books and $$35$$ pastel-sheets. Using matrices, find the total amount raised by each school.

Find the values of $$x, y$$ and $$z$$ if $$\begin{bmatrix} x+2 & 6 \\ 3 & 5z \end{bmatrix}=\begin{bmatrix} -5 & { y }^{ 2 }+y \\ 3 & -20 \end{bmatrix}$$

If $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} 7 & 11 \\ K & 23 \end{bmatrix}$$ find the value of $$k$$ .

Find the inverse of the following matrix using elementary row transformation. 
$$\begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$$

If $$A = \left[ \begin{array} { l l l } { a ^ { 2 } } & { a b } & { a c } \\ { a b } & { b ^ { 2 } } & { b c } \\ { a c } & { b c } & { c ^ { 2 } } \end{array} \right]$$ and $$a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = 1$$ then $$A ^ { 2 } =$$

If A and B are symmetric matrices, then show that AB is symmetric iff AB = BA i.e. A and B commute.

Find a matrix X such that $$2A+B+X=O$$ , where $$A=\begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix}$$ and, $$B=\begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix}$$ .

Find the value of $$X$$ and $$Y$$ if
$$2X+3Y=\begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix}$$ and $$3X+2Y=\begin{bmatrix} 2 & -2 \\ -1 & 5 \end{bmatrix}$$

Find $$x$$ and $$y$$ , if
$$\left( \begin{matrix} |x| & 2 \\ 5 & |y-2| \end{matrix} \right) = \left( \begin{matrix} <3 & 2 \\ 5 & <4 \end{matrix} \right)$$

If $$A=  \begin{bmatrix} 0 & i \\ i & 1 \end{bmatrix}$$ and $$B=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ , find the value of $$|A|+|B|$$ .

If $$A=\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}, B=\begin{bmatrix} 2 \\ 1 \end{bmatrix}$$ , then find the matrix $$X$$ such that $$A^{-1}X=B$$

$$A=\begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix}B=\begin{bmatrix} 0 & 4 \\ -1 & 7 \end{bmatrix}C=\begin{bmatrix} 1 & 0 \\ -1 & 4 \end{bmatrix}$$
find $$AC+B^{2}-10C$$ .

Find the value of $$p$$ and $$q$$ such that $$A^2 + pI = qA$$ , where $$A = \begin{bmatrix} 3 & 1 \\ 7 & 5 \end{bmatrix}$$ .

find the inverse of the matrix using row column transformations
$$\left | \begin{matrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{matrix}\right |$$

$$A=\left[ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right] ,$$ then the rank of $$AA^T$$ is

P- symmetric matrix, Q- skew symmetric matrix, P and Q are square matrice of same order, then what type of a matrix is PQ-QP

Solve the following by Matrix method
(i) 2x + 3y = 0
    2x + y - 2z = 2
    2x - z = 5

If $$A = \left[ \begin{array} { c c } { i } & { 0 } \\ { 0 } & { - i } \end{array} \right]$$ then show that $$A ^ { 2 } = - I$$

Evaluate : $$\left[ a\quad b \right] \left[ _{ d }^{ c } \right] +\left[ a\quad b\quad c\quad d \right] \left[ \underset { \underset { d }{ c }  }{ \overset { a }{ b }  }  \right]$$

If $$A=\begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$ then find $$A^{4}$$ .

Using elementary transformation find the inverse of $$A=\begin{pmatrix} 1 & 3 \\ 2 & 7 \end{pmatrix}$$

Solve the following system of equations by using Matrix inversion method. 
$$2x-y+3z=9,x+y+z=6,x-y+z=2$$

Show that the matrix $$A=\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$ satisfies the equation $$A^ {2}-4A+I=0$$ , where $$I$$ is $$2\times 2$$ identity matrix and $$0$$ is $$2\times 2$$ zero matrix. Using this equation,find $$A^ {-1}$$ .

Find $$x$$ if $$[1\ x\ 1] \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{bmatrix} \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}=0$$

If $$A=\begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}$$ and $$B=\begin{bmatrix} -3 & 1 \\ 4 & -2 \end{bmatrix}$$ then find $$A-B$$ .

If A = $$\left[ \begin{matrix} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] and\quad B=\left[ \begin{matrix} 1 & -2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \quad and$$ $${ AB=I}_{ 3 },\quad then\quad x+y\quad is\quad equal\quad to$$

Find the inverse of the following matrice by elementary transformation :

$$\begin {bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end {bmatrix}$$     

Find the adjoint of the matrix A = $$\begin {bmatrix} 1 & 4 & 3 \\  4 & 2 & 1 \\ 3 & 2 & 2 \end {bmatrix}$$ .

Find the cofactor matrix of $$\begin{bmatrix} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{bmatrix}$$

If $$A=\left[ \begin{matrix} 3  & 7 \\ 2 & 5 \end{matrix} \right]$$ and $$B=\left[ \begin{matrix} -3  & 2 \\ 4 & -1 \end{matrix} \right]$$ find the $$A+B$$

Determine the value of $$(x+y)$$ if $$\left[ \begin{matrix} 2x+y & 4x \\ 5x-7 & 4x \end{matrix} \right] =\left[ \begin{matrix} 7 & 7y-12 \\ y & x+6 \end{matrix} \right]$$

Find the value of x,y, z $$\left[ \begin{matrix} x+2 & 6 \\ 3 & { 5 }{ z } \end{matrix} \right] =\left[ \begin{matrix} 3 & { y }^{ 2 }+4 \\ 3 & 20 \end{matrix} \right]$$  

If $$A = \left[ \begin{array} { c c c } { - 1 } & { 0 }  & 2\\ { 2 } & { 3 } & { 4 } \\ { 0 } & { 1 } & { 2 } \end{array} \right] \quad B = \left[ \begin{array} { c c c } { 2 } & { 2 } & { - 4 } \\ { - 4 } & { 2 } & { - 4 } \\ { 2 } & { - 1 } & { 5 } \end{array} \right] ,$$ Find $$B+A$$

The Cartesian product of $$\{2,4\} \{3,3\} \{0, 1\}$$ is ?

Given $$A=\begin{pmatrix} 3 & -2 \\ -1 & 4 \end{pmatrix}\ B=\begin{pmatrix} 6 \\ 1 \end{pmatrix}\ C=\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$ and $$D=\begin{pmatrix} 2 \\ 2 \end{pmatrix}$$ .
Find $$AB+2C-4D$$

If $$A=\left[ \begin{matrix} 3  & 7 \\ 2 & 5 \end{matrix} \right]$$ and $$B=\left[ \begin{matrix} -3  & 2 \\ 4 & -1 \end{matrix} \right]$$ find the matrix $$C$$ if  $$2C=A+B$$

If $$A=\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$$ and $$B=\left[ \begin{matrix} p & q \\ r & s \end{matrix} \right]$$ then find $$AB$$

If $$A=\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$$ and $$B=\left[ \begin{matrix} k & l \\ m & n \end{matrix} \right]$$ then find $$AB$$

If $$A=\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right],\,\,\,B=\left[ \begin{matrix} p & q \\ r & s \end{matrix} \right]$$   and $$C=\left[ \begin{matrix} k & l \\ m & n \end{matrix} \right]$$ then find $$AB+AC$$

If $$3A - 2B =\begin{pmatrix}    1  & -2  \\    3  & 0  \end{pmatrix}$$ and $$2A - 3B =\begin{pmatrix}    -3      & 3  \\    1       & -1  \end{pmatrix}$$ then find $$B$$  

Let M = [1 -2] and N = $$\left[ \begin{matrix} 2 & 1 \\ -1 & 2 \end{matrix} \right] .$$
show that MN exists and write its order.