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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:
Report Question
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
=
?
Report Question
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
Report Question
0%
[
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
Report Question
0%
[
1
1
4
2
]
0%
[
1
4
1
2
]
0%
[
2
4
1
1
]
0%
[
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
=
[
1
1
4
2
]
Hence, option A is correct.
If
(
1
2
2
1
)
(
x
y
)
=
(
2
4
)
, then the values of
x
and
y
respectively, are
Report Question
0%
2
,
0
0%
0
,
2
0%
0
,
−
2
0%
1
,
1
Explanation
[
1
2
2
1
]
[
x
y
]
=
[
2
4
]
⇒
[
x
2
y
2
x
y
]
=
[
2
4
]
⇒
x
+
2
y
=
2
⟶
(
1
)
&
2
x
+
y
=
4
⟶
(
2
)
Multiply
(
1
)
by 2,$$
⇒
2
x
+
4
y
=
4
⟶
(
3
)
Solving
(
2
)
&
(
3
)
,
[
(
2
)
−
(
3
)
]
,
we get
⇒
−
3
y
=
0
⇒
y
=
0
Put
y
=
0
in
(
1
)
,
we get
⇒
x
+
0
=
2
⇒
x
=
2
⇒
y
=
0
Hence, the answer is
2
,
0.
If
[
5
x
1
]
[
2
−
1
3
]
=
(
20
)
, then the value of
x
is
Report Question
0%
7
0%
−
7
0%
1
7
0%
0
Explanation
Given expression:
[
5
x
1
]
1
×
3
×
[
2
−
1
3
]
3
×
1
=
(
20
)
1
×
1
⇒
(
10
+
(
−
x
)
+
3
)
1
×
1
=
(
20
)
1
×
1
⇒
13
−
x
=
20
[by equality of matrix]
⇒
x
=
−
7
If
A
=
[
7
2
1
3
]
and
A
+
B
=
[
−
1
0
2
−
4
]
, then the matrix
B
=
Report Question
0%
[
1
0
0
1
]
0%
[
6
2
3
−
1
]
0%
[
−
8
−
2
1
−
7
]
0%
[
8
2
−
1
7
]
Explanation
Given,
A
=
[
7
2
1
3
]
Also,
A
+
B
=
[
−
1
0
2
−
4
]
⇒
B
=
[
−
1
0
2
−
4
]
−
[
7
2
1
3
]
⇒
B
=
[
−
1
−
7
0
−
2
2
−
1
−
4
−
3
]
⇒
B
=
[
−
8
−
2
1
−
7
]
⇒
B
=
[
−
8
−
2
1
−
7
]
Choose the correct statement related to the matrices
A
=
[
1
0
0
1
]
and
B
=
[
0
1
1
0
]
Report Question
0%
A
3
=
A
,
B
3
≠
B
0%
A
3
≠
A
,
B
3
=
B
0%
A
3
=
A
,
B
3
=
B
0%
A
3
≠
A
,
B
3
≠
B
Explanation
Clearly,
A
is an identity matrix
Now,
A
3
=
[
1
0
0
1
]
[
1
0
0
1
]
[
1
0
0
1
]
=
[
1
0
0
1
]
[
1
0
0
1
]
=
[
1
0
0
1
]
=
A
⟹
A
3
=
A
Also,
B
3
=
[
0
1
1
0
]
[
0
1
1
0
]
[
0
1
1
0
]
=
[
1
0
0
1
]
[
0
1
1
0
]
=
[
0
1
1
0
]
=
B
⟹
B
3
=
B
Hence, option C is correct.
If
A
=
(
4
−
2
6
−
3
)
, then
A
2
is
Report Question
0%
(
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
=
A
.
A
=
[
4
−
2
6
−
3
]
[
4
−
2
6
−
3
]
=
[
16
−
12
−
8
+
6
24
−
18
−
12
+
9
]
=
[
4
−
2
6
−
3
]
Hence, the answer is
[
4
−
2
6
−
3
]
.
(
−
1
0
0
1
)
(
a
b
c
d
)
=
(
1
0
0
−
1
)
, then the values of
a
,
b
,
c
and
d
respectively are
Report Question
0%
−
1
,
0
,
0
,
−
1
0%
1
,
0
,
0
,
1
0%
−
1
,
0
,
1
,
0
0%
1
,
0
,
0
,
0
Explanation
[
−
1
0
0
1
]
[
a
b
c
d
]
=
[
1
0
0
−
1
]
⇒
[
−
a
−
b
c
d
]
=
[
1
0
0
−
1
]
On comparing both sides, corresponding columns,
⇒
−
a
=
1
,
−
b
=
0
and
c
=
0
a
=
−
1
,
b
=
0
and
d
=
−
1
Hence, the answer is
−
1
,
0
,
0
,
−
1.
If
(
a
3
1
2
)
(
2
−
1
)
=
(
5
0
)
, then the value of
a
is
Report Question
0%
8
0%
4
0%
2
0%
11
Explanation
[
a
3
1
2
]
[
2
−
1
]
=
[
5
0
]
⇒
[
2
a
−
3
0
]
=
[
5
0
]
⇒
2
a
−
3
=
5
⇒
2
a
=
8
⇒
a
=
4
If
A
=
[
1
−
2
3
]
and
B
=
[
−
1
2
−
3
]
, then
A
+
B
is
Report Question
0%
[
0
0
0
]
0%
[
0
0
0
]
0%
[
−
1
4
]
0%
not defined
Explanation
Given,
A
=
[
1
−
2
3
]
and
B
=
[
1
−
2
3
]
Here
A
+
B
is not possible because the order of the matrices is not same.
Hence, the answer is not defined.
If matrix
A
=
[
1
2
4
3
]
such that
A
x
=
I
, then
x
=
................
Report Question
0%
1
5
[
1
3
2
−
1
]
0%
1
5
[
4
2
4
−
1
]
0%
1
5
[
−
3
2
4
−
1
]
0%
1
5
[
−
1
2
−
1
4
]
Explanation
Given :
A
=
[
1
2
4
3
]
such that
A
x
=
I
...
(
i
)
Since,
A
and
I
are of order
2
×
2
. So,
x
will be a matrix of order
2
×
2
Let
x
=
[
a
b
c
d
]
From
(
i
)
, we get
A
x
=
I
⟹
[
1
2
4
3
]
[
a
b
c
d
]
=
[
1
0
0
1
]
⟹
[
a
+
2
c
b
+
2
d
4
a
+
3
c
4
b
+
3
d
]
=
[
1
0
0
1
]
⟹
a
+
2
c
=
1
......
(
i
i
)
,
4
a
+
3
c
=
0
........
(
i
i
i
)
b
+
2
d
=
0
.......
(
i
v
)
and
4
b
+
3
d
=
1
.......
(
v
)
Solving
(
i
i
)
and
(
i
i
i
)
simultaneously we get
a
=
−
3
5
and
c
=
4
5
Then solving
(
i
v
)
and
(
v
)
simultaneously we get
b
=
2
5
and
d
=
−
1
5
Substituting all these values in
x
, we get
x
=
[
−
3
5
2
5
4
5
−
1
5
]
∴
x
=
1
5
[
−
3
2
4
−
1
]
The matrix
A
=
[
0
1
−
1
−
1
0
1
1
−
1
0
]
is a :
Report Question
0%
Diagonal matrix
0%
Symmetric matrix
0%
Skew-symmetric matrix
0%
Identity matrix
Explanation
The matrix
A
=
[
0
1
−
1
−
1
0
1
1
−
1
0
]
is a
A
T
=
[
0
−
1
1
1
0
−
1
−
1
1
0
]
H
e
n
c
e
−
A
T
=
[
0
1
−
1
−
1
0
1
1
−
1
0
]
A
=
−
A
T
The matrix A is skew symmetric matrix
The inverse of a diagonal matrix is a :
Report Question
0%
Symmetric matrix
0%
Skew-symmetric matrix
0%
Diagonal matrix
0%
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
[
3
−
1
0
6
]
[
3
x
1
]
+
[
−
2
x
3
]
=
[
8
9
]
, then the value of
x
is
Report Question
0%
−
3
8
0%
7
0%
−
2
9
0%
None of these
Explanation
Given,
[
3
−
1
0
6
]
[
3
x
1
]
+
[
−
2
x
3
]
=
[
8
9
]
LHS
=
[
3
−
1
0
6
]
[
3
x
1
]
+
[
−
2
x
3
]
=
[
9
x
+
(
−
1
)
0
+
6
]
+
[
−
2
x
3
]
=
[
9
x
−
1
6
]
+
[
−
2
x
3
]
=
[
9
x
−
1
−
2
x
6
+
3
]
=
[
7
x
−
1
9
]
Now,
[
7
x
−
1
9
]
=
[
8
9
]
∴
7
x
−
1
=
8
⇒
7
x
=
9
⇒
x
=
9
7
.
Let
A
=
[
x
+
y
y
2
x
x
−
y
]
,
B
=
[
2
−
1
]
and
C
=
[
3
2
]
.
If
A
B
=
C
, then
A
2
is equal to
Report Question
0%
[
6
−
10
4
26
]
0%
[
−
10
5
4
24
]
0%
[
−
5
−
6
−
4
−
20
]
0%
None of these.
Explanation
A
=
[
x
+
y
y
2
x
x
−
y
]
,
B
=
[
2
−
1
]
and
C
=
[
3
2
]
Since,
A
B
=
C
⟹
[
x
+
y
y
2
x
x
−
y
]
[
2
−
1
]
=
[
3
2
]
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
[
1
3
0
1
]
A
=
[
1
1
0
−
1
]
, then what is equal to A?
Report Question
0%
[
1
4
0
−
1
]
0%
[
1
4
0
1
]
0%
[
−
1
4
0
−
1
]
0%
[
1
−
4
0
−
1
]
Explanation
Given
[
1
3
0
1
]
A
=
[
1
1
0
−
1
]
Let
A
=
[
α
β
γ
δ
]
[
1
3
0
1
]
[
α
β
γ
δ
]
=
[
1
1
0
−
1
]
[
α
+
3
γ
β
+
4
δ
γ
δ
]
=
[
1
1
0
−
1
]
Two matrices are equal if corresponding elements are equal
⇒
α
+
3
γ
=
1
,
γ
=
0
,
β
+
3
δ
=
1
and
δ
=
−
1
⇒
α
=
1
,
β
=
4
⇒
A
=
[
1
4
0
−
1
]
If
A
=
[
1
1
−
1
2
−
3
4
3
−
2
3
]
and
B
=
[
−
1
−
2
−
1
6
12
6
5
10
5
]
, 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 :
Report Question
0%
1 only
0%
2 only
0%
Both 1 and 2
0%
Neither 1 nor 2
Explanation
We have,
A
=
[
1
1
−
1
2
−
3
4
3
−
2
3
]
and
B
=
[
−
1
−
2
−
1
6
12
6
5
10
5
]
A
B
=
[
−
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
]
=
[
0
0
0
0
0
0
0
0
0
]
Hence,
A
B
is a null matrix.
and
A
B
≠
B
A
Hence, B is the correct option.
If
(
2
3
4
1
)
×
(
5
−
2
−
3
1
)
=
(
1
−
1
17
λ
)
then what is
λ
equal to?
Report Question
0%
7
0%
−
7
0%
9
0%
−
9
Explanation
GIven,
(
2
3
4
1
)
×
(
5
−
2
−
3
1
)
=
(
1
−
1
17
λ
)
Now,
(
2
3
4
1
)
×
(
5
−
2
−
3
1
)
=
(
10
−
9
−
4
+
3
20
−
3
−
8
+
1
)
=
(
1
−
1
17
−
7
)
So comparing both, we get
λ
=
−
7
If
A
=
(
1
3
4
5
)
then
A
−
1
equals
Report Question
0%
1
7
(
A
+
6
I
)
0%
1
7
(
A
−
6
I
)
0%
1
7
(
6
I
−
A
)
0%
None of these
Explanation
A
=
(
1
3
4
5
)
for every square matrix
A
, the characterisitc of
A
is given
|
A
−
λ
I
|
=
0
⇒
|
1
−
λ
3
4
5
−
λ
|
=
0
⇒
λ
2
−
6
λ
−
7
=
0
⇒
A
2
−
6
A
−
7
I
=
0
⇒
7
I
=
A
2
−
6
A
⇒
A
−
1
=
1
/
7
(
A
−
6
I
)
If
[
1
x
1
]
[
1
3
2
0
5
1
0
2
0
]
[
1
1
x
]
=
0
, then the values of
x
are:
Report Question
0%
1
,
5
0%
−
1
,
−
5
0%
1
,
6
0%
−
1
,
−
6
0%
3
,
3
Explanation
Since,
[
1
x
1
]
[
1
3
2
0
5
1
0
2
0
]
[
1
1
x
]
=
0
⇒
[
1
3
+
5
x
+
2
2
+
x
+
0
]
[
1
1
x
]
=
0
⇒
[
1
5
+
5
x
2
+
x
]
[
1
1
x
]
=
0
⇒
[
1
+
5
+
5
x
+
2
x
+
x
2
]
=
0
⇒
x
2
+
7
x
+
6
=
0
⇒
x
2
+
6
x
+
x
+
6
=
0
⇒
(
x
+
1
)
(
x
+
6
)
=
0
∴
x
=
−
1
,
−
6
If
A
=
[
2
3
−
1
2
]
, then
A
3
+
3
A
2
−
4
A
+
1
is equal to
Report Question
0%
[
1
1
1
0
]
0%
[
−
14
51
−
17
−
14
]
0%
[
−
14
−
51
−
17
−
14
]
0%
[
−
1
−
1
−
1
0
]
Explanation
A
=
(
2
3
−
1
2
)
∴
A
2
=
(
1
12
−
4
1
)
,
A
3
=
(
−
10
27
−
9
−
10
)
∴
A
3
+
3
A
2
−
4
A
+
t
=
(
−
10
27
−
9
−
10
)
+
(
3
36
−
12
3
)
+
(
−
8
−
12
4
−
8
)
+
(
1
0
0
1
)
=
(
−
14
51
−
17
−
14
)
Hence (b) is correct choice.
If
A
=
(
3
1
−
9
−
3
)
then
(
1
+
2
A
+
3
A
2
+
.
.
.
.
∞
)
−
1
equals
Report Question
0%
(
−
5
−
2
18
7
)
0%
(
−
5
18
−
2
7
)
0%
(
7
−
2
18
−
5
)
0%
None of these
Explanation
A
=
(
3
1
−
9
−
3
)
∴
A
2
=
(
3
1
−
9
−
3
)
(
3
1
−
9
−
3
)
=
(
0
0
0
0
)
∴
1
+
2
A
+
3
A
2
+
.
.
.
.
∞
=
1
+
2
A
=
(
1
0
0
1
)
+
(
6
2
−
18
−
6
)
=
(
7
2
−
18
−
5
)
∴
(
1
+
2
A
+
3
A
2
+
.
.
.
.
∞
)
−
1
=
(
1
+
2
A
)
−
1
=
(
7
2
−
18
−
5
)
−
1
=
(
−
5
−
2
18
7
)
If
[
α
β
γ
−
α
]
to the square is two rowed unit matrix, then
α
,
β
,
γ
should satisfy the relation
Report Question
0%
1
+
α
2
+
β
γ
=
0
0%
1
−
α
2
−
β
γ
=
0
0%
1
−
α
2
+
β
γ
=
0
0%
α
2
+
β
γ
−
1
=
0
Explanation
since
[
α
β
γ
−
α
]
is a square root of
I
2
i.e., two rowed unit matrix
∴
[
α
β
γ
−
α
]
2
=
[
1
0
0
1
]
⇒
[
1
0
0
1
]
=
[
α
β
γ
−
α
]
[
α
β
γ
−
α
]
⇒
[
1
0
0
1
]
=
[
α
2
+
β
γ
0
0
α
2
+
β
γ
]
∴
α
2
+
β
γ
=
1
⇒
1
−
α
2
−
β
γ
=
0
If
A
=
[
α
0
1
1
]
and
B
=
[
1
0
5
1
]
, then value of
α
for which
A
2
=
B
, is
Report Question
0%
1
0%
−
1
0%
4
0%
No real value
Explanation
Given :
A
=
[
α
0
1
1
]
and
B
=
[
1
0
5
1
]
A
2
=
[
α
0
1
1
]
[
α
0
1
1
]
=
[
α
2
0
α
+
1
1
]
Now
A
2
=
B
[
α
2
0
α
+
1
1
]
=
[
1
0
5
1
]
⇒
α
2
=
1
and
α
+
1
=
5
Clearly, no real value of
α
exist.
If
A
=
[
1
3
3
4
]
and
A
2
−
k
A
−
5
I
2
=
0
, then the value of
k
is
Report Question
0%
3
0%
5
0%
7
0%
−
7
Explanation
Given,
A
=
[
1
3
3
4
]
Now,
A
2
−
5
I
2
=
[
1
3
3
4
]
[
1
3
3
4
]
−
5
[
1
0
0
1
]
=
[
1
+
9
3
+
12
3
+
12
9
+
16
]
−
[
5
0
0
5
]
=
[
5
15
15
20
]
=
5
[
1
3
3
4
]
⇒
A
2
−
5
I
2
=
5
A
But it is given that
A
2
−
5
I
2
=
k
A
k
=
5
.
If
A
=
[
1
−
3
2
k
]
and
A
2
−
4
A
+
10
I
=
A
, then
k
is equal to
Report Question
0%
0
0%
−
4
0%
4
and not
1
0%
1
or
4
Explanation
Given,
A
=
[
1
−
3
2
k
]
and
A
2
−
4
A
+
10
I
=
A
⇒
[
1
−
3
2
k
]
[
1
−
3
2
k
]
−
4
[
1
−
3
2
k
]
+
10
[
1
0
0
1
]
=
[
1
−
3
2
k
]
⇒
[
−
5
−
3
−
3
k
2
+
2
k
−
6
+
k
2
]
−
[
4
−
12
8
4
k
]
+
[
10
0
0
10
]
=
[
1
−
3
2
k
]
⇒
[
1
9
−
3
k
−
6
+
2
k
4
+
k
2
−
4
k
]
=
[
1
−
3
2
k
]
⇒
9
−
3
k
=
−
3
,
−
6
+
2
k
=
2
....(i)
and
4
+
k
2
−
4
k
=
k
k
2
−
5
k
+
4
=
0
⇒
(
k
−
4
)
(
k
−
1
)
=
0
⇒
k
=
4
,
1
But
k
=
1
is not satisfied the equation (i).
If
A
=
|
5
x
−
2
2
x
+
3
x
+
1
|
is symmetric, then
x
=
_____
Report Question
0%
4
0%
5
0%
−
5
0%
−
4
Explanation
A=
|
5
x
−
2
2
x
−
3
x
+
1
|
⟹
A
T
=
|
5
2
x
−
3
x
−
2
x
+
1
|
Since
A
is symmetric,
⟹
A
=
A
T
⟹
x
−
2
=
2
x
+
3
⟹
x
=
−
5
If
A
is a non zero square matrix of order
n
with
d
e
t
(
I
+
A
)
≠
0
, and
A
3
=
0
, where
I
,
O
are unit and null matrices of order
n
×
n
respectively, then
(
I
+
A
)
−
1
=
Report Question
0%
I
−
A
+
A
2
0%
I
+
A
+
A
2
0%
I
+
A
2
0%
I
+
A
Explanation
d
e
t
(
I
+
A
)
≠
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
−
I
A
+
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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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