CBSE Questions for Class 12 Commerce Maths Matrices Quiz 6 - MCQExams.com

Given $$A, B, C$$ are three matrices such that 
$$A = \begin{bmatrix}x & y & z\end{bmatrix}$$, $$B = \begin{bmatrix} a  & h & g \\ h & b & f \\ g & f & c\end{bmatrix}$$, $$C = \begin{bmatrix}x \\ y \\ z\end{bmatrix}.$$ Evaluate $$ABC$$.
  • $$ABC = \begin{bmatrix}ax^2 + by^2 + cz^2 + 2hxy + 2gzx + 2fyz\end{bmatrix}$$
  • $$ABC = \begin{bmatrix}ax^2 - by^2 + cz^2 + 2hxy + 2gzx + 2fyz\end{bmatrix}$$
  • $$ABC = \begin{bmatrix}ax^2 - by^2 + cz^2 - 2hxy + 2gzx + 2fyz\end{bmatrix}$$
  • $$ABC = \begin{bmatrix}ax^2 + by^2 + cz^2 - 2hxy + 2gzx - 2fyz\end{bmatrix}$$
If $$A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}$$ and $$B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}$$. Find $$AB$$ and show that $$AB \ne BA$$
  • $$AB = \begin{bmatrix} 0 & -4 \\ 10 & -3 \end{bmatrix}$$
  • $$AB = \begin{bmatrix} 0 & 4 \\ 10 & 3 \end{bmatrix}$$
  • $$AB = \begin{bmatrix} 0 & -4 \\ 10 & 3 \end{bmatrix}$$
  • None of these.
If $$\displaystyle \begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix} \: \begin{bmatrix} 2 \\ -2 \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$$ then $$x-y$$ is equal to
  • $$2$$
  • $$-2$$
  • $$4$$
  • $$6$$
If $$\displaystyle \left[ \begin{matrix} 1 & x & 1 \end{matrix} \right] \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix}\: \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix}=O$$ then $$x$$ is
  • $$2$$
  • $$-2$$
  • $$14$$
  • none of these
If $$\displaystyle A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \\ 0 & 6 \end{bmatrix}$$ and $$\displaystyle B=\begin{bmatrix} 5 & 4 & 6 \\ 4 & 1 & 2 \\ -5 & -1 & 1 \end{bmatrix}$$ , then
  • $$A + B$$ exists
  • $$AB$$ exists
  • $$BA$$ exists
  • none of these
If $$\displaystyle A = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix}$$ and $$\displaystyle B = \begin{bmatrix} a^2 & ab & ac \\ ba & b^2 & bc \\ ca & cb & c^2 \end{bmatrix}$$ then $$AB$$ is equal to
  • $$[0]$$
  • $$I$$
  • $$2I$$
  • none of these
If $$A = \begin{bmatrix}1 & 3 & 0 \\ -1 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix}, \space B = \begin{bmatrix}2 & 3 & 4 \\ 1 & 2 & 3 \\ -1 & 1 & 2 \end{bmatrix}$$, then $$AB$$ = 
  • $$\begin{bmatrix}5 & 9 & 13 \\ -1 & 2 & 4 \\ -2 & 2 & 4\end{bmatrix}$$
  • $$\begin{bmatrix}5 & 9 & 13 \\ -1 & -2 & 4 \\ 2 & -2 & -4\end{bmatrix}$$
  • $$\begin{bmatrix}5 & -9 & 13 \\ 1 & -2 & 4 \\ 2 & -2 & 4\end{bmatrix}$$
  • $$\begin{bmatrix}5 & -9 & 13 \\ 1 & -2 & 4 \\ -2 & 2 & 4\end{bmatrix}$$
Given $$A = \begin{bmatrix}1 & -1 \\ 2 & -1\end{bmatrix}$$, which of the following result is true?
  • $$A^2 = I$$
  • $$A^2 = -I$$
  • $$A^2 = 2I$$
  • None of these
$$A = \begin{bmatrix}4 & x+2 \\ 2x-3 & x+1\end{bmatrix}$$ is symmetric, then $$x$$ =
  • $$3$$
  • $$5$$
  • $$2$$
  • $$4$$
If $$\displaystyle A = \begin{bmatrix} 4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3 \end{bmatrix}$$ then $$\displaystyle A^2$$ is equal to
  • $$A$$
  • $$I$$
  • $$\displaystyle A^T$$
  • none of these
If $$A$$ be a matrix such that $$\displaystyle A \times  \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}$$ then $$A$$ is
  • $$\displaystyle \begin{bmatrix}

    2 & 4 \\

    1 & -1

    \end{bmatrix}$$
  • $$\displaystyle \begin{bmatrix}

    -1 & 1 \\

    4 & 2

    \end{bmatrix}$$
  • $$\displaystyle \begin{bmatrix}

    4 & 2 \\

    -1 & 1

    \end{bmatrix}$$
  • none of these
If $$\displaystyle A=\begin{bmatrix}0 &c  &-b \\-c  &0  &a \\b  &-a  &0 \end{bmatrix}$$  and $$\displaystyle B=\begin{bmatrix}a^{2} &ab  &ac \\ab  &b^{2}  &bc \\ac  &bc  &c^{2} \end{bmatrix},$$ then $$AB=$$
  • $$\displaystyle A^{3}$$
  • $$\displaystyle B^{2}$$
  • $$I$$
  • $$O$$
If $$A = \begin{pmatrix}1& 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{pmatrix}$$

If  $$A^2 - 4A =pI $$ where $$I$$ and $$O$$ are the unit matrix and the null matrix of order $$3$$ respectively. Find the value of $$p$$
  • $$p=2$$
  • $$p=3$$
  • $$p=4$$
  • $$p=5$$
Let $$\displaystyle A = \begin{bmatrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{bmatrix}$$. The only correct statement about the matrix $$A$$ is
  • $$\displaystyle A$$ is a zero matrix
  • $$\displaystyle A = \left ( -1 \right )I_{3}$$
  • $$A^2=-I$$
  • $$\displaystyle A^{2} = I$$
If $$A$$ is a square matrix then $$A-{A}'$$ is a
  • diagonal matrix
  • skew symmetric matrix
  • symmetric matrix
  • None of these
Let $$\displaystyle A=\begin{bmatrix}-1\\2\\3 \end{bmatrix}$$ and $$\displaystyle B=\begin{bmatrix} -2 & -1 & -4 \end{bmatrix},$$  then matrix $$(AB) A$$ equals
  • $$12A$$
  • $$-12A$$
  • $$4A$$
  • $$3A$$
Use the method of elementary row transformation to compute the inverse of 
$$\quad \begin{bmatrix} 1 & 2 & 5 \\ 2 & 3 & 1 \\ -1 & 1 & 1\end{bmatrix}$$
  • $$\quad A^{-1} = \begin{bmatrix}\displaystyle\frac{2}{21} & \displaystyle\frac{1}{7} & -\displaystyle\frac{13}{21} \\ -\displaystyle\frac{1}{7} & \displaystyle\frac{2}{7} & \displaystyle\frac{3}{7}\\ \displaystyle\frac{5}{21} & -\displaystyle\frac{1}{7} & -\displaystyle\frac{1}{21}\end{bmatrix}$$
  • $$\quad A^{-1} = \begin{bmatrix}\displaystyle\frac{1}{21} & \displaystyle\frac{1}{7} & -\displaystyle\frac{11}{21} \\ -\displaystyle\frac{1}{7} & \displaystyle\frac{2}{7} & \displaystyle\frac{3}{7}\\ \displaystyle\frac{5}{21} & -\displaystyle\frac{2}{7} & -\displaystyle\frac{2}{21}\end{bmatrix}$$
  • $$\quad A^{-1} = \begin{bmatrix}\displaystyle\frac{4}{21} & \displaystyle\frac{1}{7} & -\displaystyle\frac{16}{21} \\ -\displaystyle\frac{1}{7} & \displaystyle\frac{2}{7} & \displaystyle\frac{3}{7}\\ \displaystyle\frac{5}{21} & -\displaystyle\frac{2}{7} & -\displaystyle\frac{4}{21}\end{bmatrix}$$
  • $$\quad A^{-1} = \begin{bmatrix}\displaystyle\frac{4}{21} & \displaystyle\frac{2}{7} & -\displaystyle\frac{13}{21} \\ -\displaystyle\frac{1}{7} & \displaystyle\frac{2}{7} & \displaystyle\frac{3}{7}\\ \displaystyle\frac{4}{21} & -\displaystyle\frac{2}{7} & -\displaystyle\frac{1}{21}\end{bmatrix}$$
  • Both (A) & (R) are individually true & (R) is correct explanation of (A),
  • Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
  • (A)is true but (R} is false,
  • (A)is false but (R} is true.
If $$\displaystyle A=\begin{bmatrix} 1 & \frac { 1 }{ 2 }  \\ 0 & 1 \end{bmatrix}$$ then $${ A }^{ 64 }$$ is
  • $$\begin{bmatrix} 1 & 32 \\ 32 & 1 \end{bmatrix}$$
  • $$\begin{bmatrix} 1 & 0 \\ 32 & 1 \end{bmatrix}$$
  • $$\begin{bmatrix} 1 & 32 \\ 0 & 1 \end{bmatrix}$$
  • none of these
If $$A =(a_{ij})_{3\times 3}$$ is a skew symmetric matrix, then
  • $$a_{ii}=0\: \forall \: i$$
  • $$A + A^T$$ is a null matrix
  • $$|A|=0$$
  • $$A$$ is not invertible.
$$\displaystyle A=\begin{bmatrix}a &b \\b  &a
\end{bmatrix}$$ and $$\displaystyle A^{2} =\begin{bmatrix}\alpha & \beta \\\beta & \alpha
\end{bmatrix}$$ then
  • $$\displaystyle \alpha=a^{2}+b^{2},\beta =2ab$$
  • $$\displaystyle \alpha=a^{2}+b^{2},\beta =a^{2}-b^{2}$$
  • $$\displaystyle \alpha=2ab ,\beta =a^{2}+b^{2}$$
  • $$\displaystyle \alpha=a^{2}+b^{2},\beta =ab $$
If $$\omega \neq 1$$ is cube root of unity, and $$A=\begin{bmatrix} 1&\omega   &\omega ^2 \\\omega  & \omega ^2 &1 \\ \omega ^2 &1  &\omega  \end{bmatrix}$$ is
  • symmetric
  • skew symmetric
  • singular
  • orthogonal
Let $$A$$ be a symmetric matrix such that $$A^5 =0$$ and $$B=I +A + A^2 +A^3 +A^4$$, then $$B$$ is
  • symmetric
  • singular
  • non-singular
  • skew symmetric
If $$A =\begin{bmatrix}\alpha &\beta  \\\gamma  &-\alpha  \end{bmatrix}$$ is such that $$A^2 = I$$, then

  • $$1 +\alpha^2+\beta\gamma=0$$
  • $$1 -\alpha^2-\beta\gamma=0$$
  • $$1 -\alpha^2+\beta\gamma=0$$
  • $$1 +\alpha^2-\beta\gamma=0$$
If $$A$$ and $$B$$ are two matrices of the same order, then
  • $$A^2 -B^2 = (A + B) (A -B)$$
  • $$A^2=I\Leftrightarrow (A + I) (A -I) = 0$$
  • $$(A')' = A$$
  • $$A + A'$$ is symmetric
If $$A =\begin{bmatrix} ab&b^2 \\-a^2 &-ab \end{bmatrix}$$, then $$A^2$$ is equal
  • $$O$$
  • $$I$$
  • $$-I$$
  • none of these
If A is any square matrix then (1/2) $$\displaystyle \left ( A+A^{T} \right )$$ is a _____ matrix
  • symmetric
  • skew symmetric
  • scalar
  • identity
Find the value of $$x$$ and $$y$$ that satisfy the equation:
$$\begin{bmatrix} 3&-2 \\3  &0 \\2 &4 \end{bmatrix}
\begin{bmatrix} y&y \\x &x \end{bmatrix}=\begin{bmatrix} 3&3 \\3y  &3y \\10 &10 \end{bmatrix}$$
  • $$x=3/2, \:y = 2$$
  • $$x=2, \:y = 3/2$$
  • $$x=3, \:y = 2$$
  • $$x=2, \:y = 3$$
Let $$A$$ be square matrix. Then which of the following is not a symmetric matrix.
  • $$A+A'$$
  • $$AA'$$
  • $$A'A$$
  • $$A-A'$$
Say true or false:
All positive odd integral powers of skew-symmetric matrix are symmetric.
  • True
  • False
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