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CBSE Questions for Class 12 Commerce Maths Matrices Quiz 8 - MCQExams.com
CBSE
Class 12 Commerce Maths
Matrices
Quiz 8
If
A
is a scalar matrix
k
I
with scalar
k
≠
0
of order
3
, the
A
−
1
is:
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0%
1
k
2
I
0%
1
k
3
I
0%
1
k
I
0%
k
I
Explanation
If
A
is a scalar matrix with scalar
k
, then
A
=
k
I
Thus
A
−
1
=
(
k
I
)
−
1
=
1
k
I
Note: Inverse of identity matrix is Identity matrix itself
And
(
k
A
)
−
1
=
1
k
I
−
1
=
1
k
I
If
A
=
[
7
2
1
3
]
and
A
+
B
=
[
−
1
0
2
−
4
]
,
then matrix
B
=
?
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0%
[
1
0
1
1
]
0%
[
6
2
3
−
1
]
0%
[
−
8
−
2
1
−
7
]
0%
[
8
2
−
1
7
]
Explanation
A
+
B
=
[
−
1
0
2
−
4
]
A
=
[
7
2
1
3
]
As we have learnt that we can subtract two matrices of the same order.
(Here, order of both the matrix is
2
×
2
)
So, we will subtract matrix
A
from both sides of the equation.
We get,
B
=
[
−
1
0
2
−
4
]
−
[
7
2
1
3
]
B
=
[
−
1
−
7
0
−
2
2
−
1
−
4
−
3
]
(To subtract 2 matrices, we have to subtract their corresponding elements.)
B
=
[
−
8
−
2
1
−
7
]
Hence, Option C is the correct answer
If
A
=
[
4
−
2
6
−
3
]
, then
A
2
is
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[
16
4
36
9
]
0%
[
8
−
4
12
−
6
]
0%
[
−
4
2
−
6
3
]
0%
[
4
−
2
6
−
3
]
Explanation
A
=
[
4
−
2
6
−
3
]
A
2
=
[
4
−
2
6
−
3
]
×
[
4
−
2
6
−
3
]
using the multiplication properties of matrices, we get
A
2
=
[
4
×
4
+
(
−
2
)
×
6
4
×
(
−
2
)
+
(
−
2
)
×
(
−
3
)
6
×
4
+
(
−
3
)
×
6
6
×
(
−
2
)
+
(
−
3
)
×
(
−
3
)
]
A
2
=
[
4
−
2
6
−
3
]
If
A
+
B
=
[
2
3
4
5
]
and
A
=
[
1
2
0
3
]
, then matrix
B
is
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[
1
1
4
2
]
0%
[
1
4
1
2
]
0%
[
2
4
1
1
]
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[
4
2
1
1
]
Explanation
As given
A
+
B
=
[
2
3
4
5
]
and
A
=
[
1
2
0
3
]
Substitute
A
in
A
+
B
we get,
[
1
2
0
3
]
+
B
=
[
2
3
4
5
]
⟹
B
=
[
2
3
4
5
]
−
[
1
2
0
3
]
⟹
B
=
[
(
2
−
1
)
(
3
−
2
)
(
4
−
0
)
(
5
−
3
)
]
∴
B=\begin{bmatrix} 1 & 1 \\ 4 & 2 \end{bmatrix}
Hence, option A is correct.
If
\bigl(\begin{smallmatrix} 1& 2\\ 2 & 1\end{smallmatrix}\bigr) \bigl(\begin{smallmatrix} x \\ y \end{smallmatrix}\bigr) = \bigl(\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\bigr)
, then the values of
x
and
y
respectively, are
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2, 0
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0, 2
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0, -2
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1, 1
Explanation
\left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix} \right] \left[ \begin{matrix} x \\ y \end{matrix} \right] =\left[ \begin{matrix} 2 \\ 4 \end{matrix} \right]
\Rightarrow \left[ \begin{matrix} x & 2y \\ 2x & y \end{matrix} \right] =\left[ \begin{matrix} 2 \\ 4 \end{matrix} \right]
\Rightarrow x+2y=2\longrightarrow (1)\;\; \& \;\;2x+y=4\longrightarrow (2)
Multiply
(1)
by 2,$$
\Rightarrow 2x+4y=4\longrightarrow(3)
Solving
(2)\;\&\;(3),[(2)-(3)],
we get
\Rightarrow -3y=0
\Rightarrow y=0
Put
y=0
in
(1),
we get
\Rightarrow x+0=2
\Rightarrow x=2
\Rightarrow y=0
Hence, the answer is
2,0.
If
\begin{bmatrix} 5 & x & 1 \end{bmatrix} \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} = (20)
, then the value of
x
is
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7
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-7
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\dfrac{1}{7}
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0
Explanation
Given expression:
{ \begin{bmatrix} 5 &x&1\end{bmatrix} }_{ 1\times 3 }\times { \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} }_{ 3\times 1 }={ \begin{pmatrix} 20 \end{pmatrix} }_{ 1\times 1 }\\\Rightarrow { (10+(-x)+3) }_{ 1\times 1 }={ \begin{pmatrix} 20 \end{pmatrix} }_{ 1\times 1 }
\Rightarrow 13-x=20
[by equality of matrix]
\Rightarrow x=-7
If
A = \begin{bmatrix}7 &2 \\ 1 & 3\end{bmatrix}
and
A + B = \begin{bmatrix} -1& 0\\ 2 & -4\end{bmatrix}
, then the matrix
B =
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\left[\begin{matrix}1 &0 \\ 0 & 1\end{matrix}\right]
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\left[\begin{matrix} 6&2 \\ 3 & -1\end{matrix}\right]
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\left[\begin{matrix} -8& -2\\ 1 & -7\end{matrix}\right]
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\left[\begin{matrix} 8& 2\\ -1 & 7\end{matrix}\right]
Explanation
Given,
A=\begin{bmatrix} 7 & 2 \\ 1 & 3 \end{bmatrix}
Also,
A+B=\begin{bmatrix} -1 & 0 \\ 2 & -4 \end{bmatrix}\\ \Rightarrow B=\begin{bmatrix} -1 & 0 \\ 2 & -4 \end{bmatrix}-\begin{bmatrix} 7 & 2 \\ 1 & 3 \end{bmatrix}\\ \Rightarrow B=\begin{bmatrix} -1-7 & 0-2 \\ 2-1 & -4-3 \end{bmatrix}\\ \Rightarrow B=\begin{bmatrix} -8 & -2 \\ 1 & -7 \end{bmatrix}
\Rightarrow B=\begin{bmatrix} -8 & -2 \\ 1 & -7 \end{bmatrix}
Choose the correct statement related to the matrices
A=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}
and
B=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}
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A^3=A, B^3 \ne B
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A^3\ne A, B^3=B
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A^3=A, B^3 = B
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A^3\ne A, B^3 \ne B
Explanation
Clearly,
A
is an identity matrix
Now,
{ A }^{ 3 }=
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\quad
= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = A
\implies A^3=A
Also,
{ B}^{ 3 }= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = B
\implies B^{3}=B
Hence, option C is correct.
If
A = \bigl(\begin{smallmatrix} 4& -2\\ 6 & -3\end{smallmatrix}\bigr)
, then
A^2
is
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\bigl(\begin{smallmatrix} 16 & 4\\ 36 & 9\end{smallmatrix}\bigr)
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\bigl(\begin{smallmatrix}8 & -4\\ 12 & -6\end{smallmatrix}\bigr)
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\bigl(\begin{smallmatrix} -4& 2\\ -6 & 3\end{smallmatrix}\bigr)
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\bigl(\begin{smallmatrix} 4& -2\\ 6 & -3\end{smallmatrix}\bigr)
Explanation
A=\left[ \begin{matrix} 4 & -2 \\ 6 & -3 \end{matrix} \right]
\Rightarrow { A }^{ 2 }=A.A=\left[ \begin{matrix} 4 & -2 \\ 6 & -3 \end{matrix} \right] \left[ \begin{matrix} 4 & -2 \\ 6 & -3 \end{matrix} \right]
=\left[ \begin{matrix} 16-12 & -8+6 \\ 24-18 & -12+9 \end{matrix} \right]
=\left[ \begin{matrix} 4 & -2 \\ 6 & -3 \end{matrix} \right]
Hence, the answer is
\left[ \begin{matrix} 4 & -2 \\ 6 & -3 \end{matrix} \right] .
\bigl(\begin{smallmatrix} -1& 0\\ 0 & 1\end{smallmatrix}\bigr) \bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr) = \bigl(\begin{smallmatrix} 1& 0\\ 0 & -1\end{smallmatrix}\bigr)
, then the values of
a, b, c
and
d
respectively are
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-1, 0, 0, -1
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1, 0, 0, 1
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-1, 0, 1, 0
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1, 0, 0, 0
Explanation
\left[ \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right] \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] =\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]
\Rightarrow \left[ \begin{matrix} -a & -b \\ c & d \end{matrix} \right] =\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]
On comparing both sides, corresponding columns,
\Rightarrow -a=1,
-b=0
and
c=0
a=-1,
b=0
and
d=-1
Hence, the answer is
-1,0,0,-1.
If
\bigl(\begin{smallmatrix}a & 3\\ 1 & 2\end{smallmatrix}\bigr) \bigl(\begin{smallmatrix} 2 \\ -1 \end{smallmatrix}\bigr) = \bigl(\begin{smallmatrix} 5\\ 0 \end{smallmatrix}\bigr)
, then the value of
a
is
Report Question
0%
8
0%
4
0%
2
0%
11
Explanation
\left[ \begin{matrix} a & 3 \\ 1 & 2 \end{matrix} \right] \left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} 5 \\ 0 \end{matrix} \right]
\Rightarrow \left[ \begin{matrix} 2a-3 \\ 0 \end{matrix} \right] =\left[ \begin{matrix} 5 \\ 0 \end{matrix} \right]
\Rightarrow 2a-3=5
\Rightarrow 2a=8
\Rightarrow a=4
If
A=\left[ \begin{matrix} 1 & -2 & 3 \end{matrix} \right]
and
B=\left[ \begin{matrix} -1 \\ 2 \\ -3 \end{matrix} \right]
, then
A + B
is
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\begin{bmatrix} 0& 0 & 0 \end{bmatrix}
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\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
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\begin{bmatrix} -1& 4 \end{bmatrix}
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not defined
Explanation
Given,
A=\left[ \begin{matrix} 1 & -2 & 3 \end{matrix} \right]
and
B=\left[ \begin{matrix} 1 \\ -2 \\ 3 \end{matrix} \right]
Here
A+B
is not possible because the order of the matrices is not same.
Hence, the answer is not defined.
If matrix
A=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix}
such that
Ax=I
, then
x=
................
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\cfrac { 1 }{ 5 } \begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix}
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\cfrac { 1 }{ 5 } \begin{bmatrix} 4 & 2 \\ 4 & -1 \end{bmatrix}
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\cfrac { 1 }{ 5 } \begin{bmatrix} -3 & 2 \\ 4 & -1 \end{bmatrix}
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\cfrac { 1 }{ 5 } \begin{bmatrix} -1 & 2 \\ -1 & 4 \end{bmatrix}
Explanation
Given :
A=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix}
such that
Ax=I
...
(i)
Since,
A
and
I
are of order
2\times 2
. So,
x
will be a matrix of order
2\times 2
Let
x=\begin{bmatrix} a & b \\ c & d \end{bmatrix}
From
(i)
, we get
Ax=I
\implies \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\implies \begin{bmatrix} a+2c & b+2d \\ 4a+3c & 4b+3d \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\implies a+2c=1
......
(ii),
4a+3c=0
........
(iii)
b+2d=0
.......
(iv)
and
4b+3d=1
.......
(v)
Solving
(ii)
and
(iii)
simultaneously we get
a=-\dfrac{3}{5}
and
c=\dfrac{4}{5}
Then solving
(iv)
and
(v)
simultaneously we get
b=\dfrac{2}{5}
and
d=-\dfrac{1}{5}
Substituting all these values in
x
, we get
x=\begin{bmatrix} -\dfrac{3}{5} & \dfrac{2}{5} \\ \dfrac{4}{5} & -\dfrac{1}{5} \end{bmatrix}
\therefore x=\dfrac{1}{5} \begin{bmatrix} -3 & 2 \\ 4 & -1 \end{bmatrix}
The matrix
A = \begin{bmatrix} 0& 1 & -1\\ -1 & 0 & 1\\ 1 & -1 & 0\end{bmatrix}
is a :
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Diagonal matrix
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Symmetric matrix
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Skew-symmetric matrix
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Identity matrix
Explanation
The matrix
A=\quad \begin{bmatrix} 0 & \quad \quad 1 & \quad \quad -1 \\ -1 & \quad \quad 0 & \quad \quad 1 \\ 1 & \quad \quad -1 & \quad \quad 0 \end{bmatrix}
is a
{ \quad A }^{ T\quad }=\quad \begin{bmatrix} 0 & \quad -1 & \quad \quad 1 \\ 1 & \quad \quad 0 & \quad \quad -1 \\ -1 & \quad \quad 1 & \quad \quad 0 \end{bmatrix}
Hence \ \ \ { -A }^{ T }\quad =\quad \begin{bmatrix} 0 & \quad \quad 1 & \quad \quad -1 \\ -1 & \quad \quad 0 & \quad \quad 1 \\ 1 & \quad \quad -1 & \quad \quad 0 \end{bmatrix}\quad \\ \\ \quad A\quad =\quad { -A }^{ T }
The matrix A is skew symmetric matrix
The inverse of a diagonal matrix is a :
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Symmetric matrix
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Skew-symmetric matrix
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Diagonal matrix
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None of the above
Explanation
A diagonal matrix has elements only in it's diagonal.
So the inverse will also have all non zero elements in the diagonal.
So, it will be symmetric and will also be a diagonal matrix.
Hence, option A and C are correct
If
\begin{bmatrix} 3 & -1 \\ 0 & 6 \end{bmatrix}\begin{bmatrix} 3x \\ 1 \end{bmatrix}+\begin{bmatrix} -2x \\ 3 \end{bmatrix}=\begin{bmatrix} 8 \\ 9 \end{bmatrix}
, then the value of
x
is
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-\dfrac { 3 }{ 8 }
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7
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-\dfrac { 2 }{ 9 }
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None of these
Explanation
Given,
\begin{bmatrix} 3 & -1 \\ 0 & 6 \end{bmatrix}\begin{bmatrix} 3x \\ 1 \end{bmatrix}+\begin{bmatrix} -2x \\ 3 \end{bmatrix}=\begin{bmatrix} 8 \\ 9 \end{bmatrix}
LHS
=\begin{bmatrix} 3 & -1 \\ 0 & 6 \end{bmatrix}\begin{bmatrix} 3x \\ 1 \end{bmatrix}+\begin{bmatrix} -2x \\ 3 \end{bmatrix}
=\begin{bmatrix} 9x+\left( -1 \right) \\ 0+6 \end{bmatrix}+\begin{bmatrix} -2x \\ 3 \end{bmatrix}=\begin{bmatrix} 9x-1 \\ 6 \end{bmatrix}+\begin{bmatrix} -2x \\ 3 \end{bmatrix}
=\begin{bmatrix} 9x-1-2x \\ 6+3 \end{bmatrix}=\begin{bmatrix} 7x-1 \\ 9 \end{bmatrix}
Now,
\begin{bmatrix} 7x-1 \\ 9 \end{bmatrix}=\begin{bmatrix} 8 \\ 9 \end{bmatrix}
\therefore 7x-1=8
\Rightarrow 7x=9
\Rightarrow x=\dfrac { 9 }{ 7 }
.
Let
A = \begin{bmatrix}x + y & y\\ 2x & x - y\end{bmatrix}, B = \begin{bmatrix} 2& -1\end{bmatrix}
and
C = \begin{bmatrix} 3& 2\end{bmatrix}.
If
AB = C
, then
A^{2}
is equal to
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\begin{bmatrix} 6 & -10\\ 4 & 26 \end{bmatrix}
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\begin{bmatrix} -10 & 5\\ 4 & 24 \end{bmatrix}
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\begin{bmatrix} -5 & -6\\ -4 & -20 \end{bmatrix}
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None of these.
Explanation
A=\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}, B=\begin{bmatrix} 2 & -1 \end{bmatrix}
and
C=\begin{bmatrix} 3 & 2 \end{bmatrix}
Since,
AB=C\\ \Longrightarrow \begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}\begin{bmatrix} 2 & -1 \end{bmatrix}=\begin{bmatrix} 3 & 2 \end{bmatrix}
The multiplication of
A
and
B
can't be done because it doesn't satisfy the conditions of matrix multipication
If the matrix A is such that
\begin{bmatrix} 1 & 3\\ 0 & 1\end{bmatrix} A=\begin{bmatrix} 1 & 1 \\ 0 & -1\end{bmatrix}
, then what is equal to A?
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\begin{bmatrix} 1 & 4 \\ 0 & -1\end{bmatrix}
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\begin{bmatrix} 1 & 4 \\ 0 & 1\end{bmatrix}
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\begin{bmatrix} -1 & 4\\ 0 & -1\end{bmatrix}
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\begin{bmatrix} 1 & -4 \\ 0 & -1\end{bmatrix}
Explanation
Given
\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}A=\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}
Let
A=\begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix}
\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix}=\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}
\begin{bmatrix} \alpha +3\gamma & \beta +4\delta \\ \gamma & \delta \end{bmatrix}=\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}
Two matrices are equal if corresponding elements are equal
\Rightarrow \alpha +3\gamma =1, \gamma =0,
\beta +3\delta =1
and
\delta =-1
\Rightarrow \alpha =1,
\beta =4
\Rightarrow A=\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}
If
A=\begin{bmatrix}1&1&-1\\2&-3&4\\3&-2&3\end{bmatrix}
and
B=\begin{bmatrix}-1&-2&-1\\6&12&6\\5&10&5\end{bmatrix}
, then which of the following is/are correct?
A and B commute.
AB is null matrix.
Select the correct answer using the code given below :
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1 only
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2 only
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Both 1 and 2
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Neither 1 nor 2
Explanation
We have,
A=\begin{bmatrix}1&1&-1\\2&-3&4\\3&-2&3\end{bmatrix}
and
B=\begin{bmatrix}-1&-2&-1\\6&12&6\\5&10&5\end{bmatrix}
AB=\begin{bmatrix}-1+6-5&-2+12-10&-1+6-5\\-2-18+20&-4-36+40&-2-18+20\\-3-12+15&-6-24+30&-3-12+15\end{bmatrix}
=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}
Hence,
AB
is a null matrix.
and
AB\neq BA
Hence, B is the correct option.
If
\begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix}\times \begin{pmatrix} 5 & -2 \\ -3 & 1 \end{pmatrix}=\begin{pmatrix} 1 & -1 \\ 17 & \lambda \end{pmatrix}
then what is
\lambda
equal to?
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7
0%
-7
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9
0%
-9
Explanation
GIven,
\begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix}\times \begin{pmatrix} 5 & -2 \\ -3 & 1 \end{pmatrix}=\begin{pmatrix} 1 & -1 \\ 17 & \lambda \end{pmatrix}
Now,
\begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix}\times \begin{pmatrix} 5 & -2 \\ -3 & 1 \end{pmatrix}=\begin{pmatrix} 10-9 & -4+3 \\ 20-3 & -8+1 \end{pmatrix}=\begin{pmatrix} 1 & -1 \\ 17 & -7 \end{pmatrix}
So comparing both, we get
\lambda =-7
If
\quad A=\begin{pmatrix} 1 & 3 \\ 4 & 5 \end{pmatrix}
then
{ A }^{ -1 }
equals
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\cfrac { 1 }{ 7 } \left( A+6I \right)
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\cfrac { 1 }{ 7 } \left( A-6I \right)
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\cfrac { 1 }{ 7 } \left( 6I-A \right)
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None of these
Explanation
A=\begin{pmatrix} 1 & 3 \\ 4 & 5 \end{pmatrix}
for every square matrix
A
, the characterisitc of
A
is given
\left| A-\lambda I \right| =0
\Rightarrow \begin{vmatrix} 1-\lambda & 3 \\ 4 & 5-\lambda \end{vmatrix}=0
\Rightarrow { \lambda }^{ 2 }-6\lambda -7=0\quad \Rightarrow { A }^{ 2 }-6A-7I=0\Rightarrow 7I={ A }^{ 2 }-6A\quad \quad
\Rightarrow { A }^{ -1 }=1/7(A-6I)\quad
If
[1\,x\,1] \begin{bmatrix} 1&3&2 \\ 0&5&1\\0&2&0 \end{bmatrix}
\begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix}=0
, then the values of
x
are:
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1,5
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-1,-5
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1,6
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-1,-6
0%
3,3
Explanation
Since,
[1\,x\,1] \begin{bmatrix} 1&3&2 \\ 0&5&1\\0&2&0 \end{bmatrix}
\begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix}=0
\Rightarrow \begin{bmatrix} 1 & 3+5x+2 & 2+x+0 \end{bmatrix}\begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix}=0
\Rightarrow \begin{bmatrix} 1 &5+5x & 2+x \end{bmatrix}\begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix}=0
\Rightarrow \left[ 1+5+5x+2x+{ x }^{ 2 } \right] =0
\Rightarrow { x }^{ 2 }+7x+6=0
\Rightarrow x^2+6x+x+6=0
\Rightarrow (x+1)(x+6)=0
\therefore x=-1,-6
If
A = \begin{bmatrix} 2& 3\\ -1 & 2\end{bmatrix}
, then
A^{3} + 3A^{2} - 4A + 1
is equal to
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\begin{bmatrix} 1& 1\\ 1 & 0\end{bmatrix}
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\begin{bmatrix} -14& 51\\ -17 & -14\end{bmatrix}
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\begin{bmatrix} -14& -51\\ -17 & -14\end{bmatrix}
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\begin{bmatrix} -1& -1\\ -1 & 0\end{bmatrix}
Explanation
A = \begin{pmatrix} 2& 3\\ -1 & 2\end{pmatrix}
\therefore A^{2} = \begin{pmatrix} 1& 12\\ -4 & 1\end{pmatrix}, A^{3} = \begin{pmatrix} -10& 27\\ -9 & -10\end{pmatrix}
\therefore A^{3} + 3A^{2} - 4A + t = \begin{pmatrix} -10& 27\\ -9 & -10\end{pmatrix} + \begin{pmatrix} 3& 36\\ -12 & 3\end{pmatrix} + \begin{pmatrix} -8& -12\\ 4 & -8\end{pmatrix} + \begin{pmatrix} 1& 0\\ 0 & 1\end{pmatrix}
= \begin{pmatrix} -14& 51\\ -17 & -14\end{pmatrix}
Hence (b) is correct choice.
If
A=\begin{pmatrix} 3 & 1 \\ -9 & -3 \end{pmatrix}
then
{ \left( 1+2A+3{ A }^{ 2 }+....\infty \right) }^{ -1 }
equals
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\begin{pmatrix} -5 & -2 \\ 18 & 7 \end{pmatrix}
0%
\begin{pmatrix} -5 & 18 \\ -2 & 7 \end{pmatrix}
0%
\begin{pmatrix} 7 & -2 \\ 18 & -5 \end{pmatrix}
0%
None of these
Explanation
A=\begin{pmatrix} 3 & 1 \\ -9 & -3 \end{pmatrix}
\quad \therefore { A }^{ 2 }=\begin{pmatrix} 3 & 1 \\ -9 & -3 \end{pmatrix}\begin{pmatrix} 3 & 1 \\ -9 & -3 \end{pmatrix}=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
\therefore 1+2A+3{ A }^{ 2 }+....\infty =1+2A
=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}+\begin{pmatrix} 6 & 2 \\ -18 & -6 \end{pmatrix}=\begin{pmatrix} 7 & 2 \\ -18 & -5 \end{pmatrix}
\therefore (1+2A+3{ A }^{ 2 }+....\infty)^{-1}=(1+2A)^{-1}
\quad ={ \begin{pmatrix} 7 & 2 \\ -18 & -5 \end{pmatrix} }^{ -1 }=\begin{pmatrix} -5 & -2 \\ 18 & 7 \end{pmatrix}
If
\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}
to the square is two rowed unit matrix, then
\alpha ,\beta ,\gamma
should satisfy the relation
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1+{ \alpha }^{ 2 }+\beta \gamma =0
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1-{ \alpha }^{ 2 }-\beta \gamma =0
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1-{ \alpha }^{ 2 }+\beta \gamma =0
0%
{ \alpha }^{ 2 }+\beta \gamma -1=0
Explanation
since
\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}
is a square root of
{I}_{2}
i.e., two rowed unit matrix
\therefore { \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} }^{ 2 }=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\Rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}
\Rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} { \alpha }^{ 2 }+\beta \gamma & 0 \\ 0 & { \alpha }^{ 2 }+\beta \gamma \end{bmatrix}
\therefore { \alpha }^{ 2 }+\beta \gamma =1
\Rightarrow 1-{ \alpha }^{ 2 }-\beta \gamma =0
If
A=\begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}
and
B=\begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}
, then value of
\alpha
for which
{A}^{2}=B
, is
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1
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-1
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4
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No real value
Explanation
Given :
A=\begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}
and
B=\begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}
{ A }^{ 2 }=\begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}=\begin{bmatrix} { \alpha }^{ 2 } & 0 \\ \alpha +1 & 1 \end{bmatrix}
Now
{ A }^{ 2 }=B\quad
\begin{bmatrix} { \alpha }^{ 2 } & 0 \\ \alpha +1 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}
\Rightarrow { \alpha }^{ 2 }=1
and
\alpha +1=5
Clearly, no real value of
\alpha
exist.
If
A = \begin{bmatrix}1 &3 \\ 3 & 4\end{bmatrix}
and
A^{2} - kA - 5I_{2} = 0
, then the value of
k
is
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3
0%
5
0%
7
0%
-7
Explanation
Given,
A = \begin{bmatrix}1 &3 \\ 3 & 4\end{bmatrix}
Now,
A^{2} - 5I_{2} = \begin{bmatrix}1 &3 \\ 3 & 4\end{bmatrix}\begin{bmatrix}1 &3 \\ 3 & 4\end{bmatrix} - 5\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}
= \begin{bmatrix}1 + 9 & 3 + 12 \\ 3 + 12 & 9 + 16\end{bmatrix} - \begin{bmatrix}5 &0 \\ 0 & 5\end{bmatrix}
= \begin{bmatrix}5 &15 \\ 15 & 20\end{bmatrix} = 5\begin{bmatrix}1 &3 \\ 3 & 4\end{bmatrix}
\Rightarrow A^{2} - 5I_{2} = 5A
But it is given that
A^{2} - 5I_{2} = kA
k = 5
.
If
A=\begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix}
and
{ A }^{ 2 }-4A+10I=A
, then
k
is equal to
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0
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-4
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4
and not
1
0%
1
or
4
Explanation
Given,
A=\begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix}
and
{ A }^{ 2 }-4A+10I=A
\Rightarrow \begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix} \begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix} -4 \begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix} + 10 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix}
\Rightarrow \begin{bmatrix} -5 & -3-3k \\ 2+2k & -6+{ k }^{ 2 } \end{bmatrix} - \begin{bmatrix} 4 & -12 \\ 8 & 4k \end{bmatrix} + \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix} = \begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix}
\Rightarrow \begin{bmatrix} 1 & 9-3k \\ -6+2k & 4+{ k }^{ 2 }-4k \end{bmatrix} = \begin{bmatrix} 1 & -3 \\ 2 & k \end{bmatrix}
\Rightarrow 9-3k=-3, -6+2k=2
....(i)
and
4+{ k }^{ 2 }-4k=k
{ k }^{ 2 }-5k+4=0
\Rightarrow \left( k-4 \right) \left( k-1 \right) =0 \Rightarrow k=4,1
But
k=1
is not satisfied the equation (i).
If
A = \begin{vmatrix} 5 & x-2 \\ 2x+3 & x+1 \end{vmatrix}
is symmetric, then
x =
_____
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4
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5
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-5
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-4
Explanation
A=
\begin {vmatrix} 5&x-2\\2x-3&x+1\end{vmatrix}
\implies A^T=
\begin {vmatrix} 5&2x-3\\x-2&x+1\end{vmatrix}
Since
A
is symmetric,
\implies A=A^T
\implies x-2=2x+3
\implies x=-5
If
A
is a non zero square matrix of order
n
with
det\left( I+A \right) \neq 0
, and
{A}^{3}=0
, where
I,O
are unit and null matrices of order
n\times n
respectively, then
{ \left( I+A \right) }^{ -1 }=
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I-A+{ A }^{ 2 }
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I+A+{ A }^{ 2 }
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I+{ A }^{ 2 }
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I+A
Explanation
det(I+A)\neq 0
A^3=0
where
0
is null matrix,
I
is the identity matrix
A^3+I=I
[adding
I
on both sides]
(A+I)(A^2-IA+I^2)=I
[by the formula of
a^3+b^3
]
(A+I)(A^2-A+I)=I
(I+A)(I+A)^{-1}=I
[by the rule of inverse matrix]
hence
(I+A)^{-1}=(A^2-A+I)
Ans:
I-A+A^2
0:0:1
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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