CBSE Class 12 Maths –Chapter 3 Matrices- Study Materials

Matrices Class 12 MCQs Questions with Answers

Question 1.
\(\left|\begin{array}{lll}
3 & 4 & 5 \\
0 & 2 & 3 \\
0 & 0 & 7
\end{array}\right|\) = A then |A| = ?
(a) 40
(b) 50
(c) 42
(d) 15

Question 2.
The inverse of A = \(\left|\begin{array}{ll}
2 & 3 \\
5 & k
\end{array}\right|\) will not be obtained if A has the value
(a) 2
(b) \(\frac{3}{2}\)
(c) \(\frac{5}{2}\)
(d) \(\frac{15}{2}\)

Question 3.
For any unit matrix I
(a) I² = I
(b) |I| = 0
(c) |I| = 2
(d) |I| = 5

Question 4.
A matrix A = [a_ij]_m×n is said to be symmetric if
(a) a_ij = 0
(b) a_ij = a_ji
(c) a_ij = a_ij
(d) a_ij = 1

Question 5.
If A = \(\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right|\) then A² is
(a) 27 A
(b) 2 A
(c) 3 A
(d) 1

Question 6.
A matrix A = [a_ij]_m×n is said to be skew symmetric if
(a) a_ij = 0
(b) a_ij = a_ji
(c) a_ij = -a_ji
(d) a_ij = 1

Question 7.
A = [a_ij]_m×n is a square matrix if
(a) m = n
(b) m < n
(c) m > n
(d) None of these

Question 8.
If A and B are square matrices then (AB)’ =
(a) B’A’
(b) A’B’
(c) AB’
(d) A’B’

Question 9.
If A = \(\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]\) and adj A is
(a) \(\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\)
(d) \(\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right]\)

Question 10.
If \(\left[\begin{array}{cc}
1-x & 2 \\
18 & 6
\end{array}\right]\) = \(\left[\begin{array}{cc}
6 & 2 \\
18 & 6
\end{array}\right]\) then x =
(a) ±6
(b) 6
(c) -5
(d) 7

Question 11.
If \(\left|\begin{array}{ll}
x & 8 \\
3 & 3
\end{array}\right|\) = 0, the value of x is
(a) 3
(b) 8
(c) 24
(d) 0

Question 12.
If A = \(\left[\begin{array}{cc}
i & 0 \\
0 & i
\end{array}\right]\) then A² =
(a) \(\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]\)
(d) \(\left[\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}\right]\)

Question 13.
Let A be a non-singular matrix of the order 2 × 2 then |A^-1|=
(a) |A|
(b) \(\frac{1}{|A|}\)
(c) 0
(d) 1

Question 14.
If A = \(\left[\begin{array}{cc}
1 & 2 \\
2 & 1
\end{array}\right]\) then adj A =
(a) \(\left[\begin{array}{cc}
1 & -2 \\
-2 & 1
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
2 & 1 \\
1 & 1
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
1 & -2 \\
-2 & -1
\end{array}\right]\)
(d) \(\left[\begin{array}{cc}
-1 & 2 \\
-2 & -1
\end{array}\right]\)

Question 15.
If A = \(\left[\begin{array}{cc}
1 & 1 \\
0 & 1
\end{array}\right]\) B = \(\left[\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right]\) then AB =
(a) \(\left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
1 & 1 \\
1 & 0
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]\)
(d) 10

Question 16.
If \(\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
a & b & -1
\end{array}\right]\) then A² =
(a) a unit matrix
(b) A
(c) a null matrix
(d) -A

Question 17.
If A = \(\left[\begin{array}{cc}
α & 0 \\
1 & 1
\end{array}\right]\) B = \(\left[\begin{array}{cc}
1 & 0 \\
5 & 1
\end{array}\right]\) where A² = B then the value of α is
(a) 1
(b) -1
(c) 4
(d) we cant calculate the value of α

Question 18.
If A = \(\left[\begin{array}{cc}
1 & 2 \\
3 & 4
\end{array}\right]\) then
(a) |A| = 0
(b) A^-1 exists
(c) A^-1 does not exist
(d) None of these

Question 19.
If A = \(\left[\begin{array}{cc}
2x & 5 \\
8 & x
\end{array}\right]\) = \(\left[\begin{array}{cc}
6 & -2 \\
7 & 3
\end{array}\right]\) then the value of x is
(a) 3
(b) ±3
(c) ±6
(d) 6

Question 20.
Let A = \(\left[\begin{array}{cc}
1 & -1 \\
2 & 3
\end{array}\right]\) then
(a) A^-1 = \(\left[\begin{array}{cc}
\frac{3}{5} & \frac{1}{5} \\
\frac{-2}{5} & \frac{1}{5}
\end{array}\right]\)
(b) |A| = 0
(c) |A| = 5
(d) A² = 1

Question 21.
If A = \( \left[\begin{array}{ccc}
2 & \lambda & -3 \\
0 & 2 & 5 \\
1 & 1 & 3
\end{array}\right]\) yhen A^-1 exists if
(a) λ = 2
(b) λ ≠ 2
(c) λ ≠ -2
(d) none of these

Question 22.
If A = \(\left[\begin{array}{cc}
α & 2 \\
2 & α
\end{array}\right]\) and |A³| = 25 then α is
(a) ±3
(b) ±2
(c) ±5
(d) 0

Question 23.
A² – A + I = 0 then the inverse of A
(a) A
(b) A + I
(c) I – A
(d) A – I

Question 24.
If A = \(\left[\begin{array}{cc}
2 & 3 \\
1 & -4
\end{array}\right]\) and B = \(\left[\begin{array}{cc}
1 & -2 \\
-1 & 3
\end{array}\right]\) then find (AB)^-1
(a) \(\frac{1}{11}\) \(\left[\begin{array}{cc}
14 & 5 \\
5 & 1
\end{array}\right]\)
(b) \(\frac{1}{11}\) \(\left[\begin{array}{cc}
14 & -5 \\
-5 & 1
\end{array}\right]\)
(c) \(\frac{1}{11}\) \(\left[\begin{array}{cc}
1 & 5 \\
5 & 14
\end{array}\right]\)
(d) \(\frac{1}{11}\) \(\left[\begin{array}{cc}
1 & -5 \\
-5 & 14
\end{array}\right]\)

Question 25.
If A = \(\left[\begin{array}{cc}
3 & 1 \\
-1 & 2
\end{array}\right]\) then A² – 5A – 7I is
(a) zero matrix
(b) a diagonal matrix
(c) identity matrix
(d) None of these

Question 26.
If A = \(\left[\begin{array}{cc}
\cos x & -\sin x \\
\sin x & \cos x
\end{array}\right]\) then A + A^T = I if the value of x is
(a) \(\frac{π}{6}\)
(b) \(\frac{π}{3}\)
(c) π
(d) 0

Question 27.
If \(\left[\begin{array}{cc}
x+y & y \\
2x & x-y
\end{array}\right]\) \(\left[\begin{array}{c}
2 \\
-1
\end{array}\right]\) \(\left[\begin{array}{c}
3 \\
2
\end{array}\right]\) then xy equal to
(a) -5
(b) -4
(c) 4
(d) 5

Question 28.
If A = \(\left[\begin{array}{cc}
1 & 2 \\
4 & 2
\end{array}\right]\) then |2A| =
(a) 2|A|
(b) 4|A|
(c) 8|A|
(d) None of these

Question 29.
If A = \(\left[\begin{array}{cc}
a & b \\
c & d
\end{array}\right]\) then A² is equal to
(a) \(\left[\begin{array}{cc}
a^{2} & b^{2} \\
c^{2} & d^{2}
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
b^{2}+bc & ab+bd \\
ac+dc & dc+d^{2}
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
a^{3} & b^{3} \\
c^{3} & d^{3}
\end{array}\right]\)
(d) None of these

Question 30.
\(\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\) is inverse of
(a) \(\left[\begin{array}{cc}
-\cos \theta & -\sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
\cos \theta & \sin \theta \\
\sin \theta & -\cos \theta
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\)
(d) None of these

Question 31.
A = \(\left[\begin{array}{cc}
a & b \\
b & a
\end{array}\right]\) and A² = \(\left[\begin{array}{cc}
α & β \\
β & α
\end{array}\right]\) then
(a) α = a² + b², β = ab
(b) α = a² + b², β = 2ab
(c) α = a² + b², β = a² – b²
(d) α = 2ab, β = a² + b²

Question 32.
The matrix \(\left[\begin{array}{ccc}
2 & -1 & 4 \\
1 & 0 & -5 \\
-4 & 5 & 7
\end{array}\right]\) is
(a) a symmetric matix
(b) a skew-sybtmetric matrix
(c) a diagonal matrix
(d) None of these

Question 33.
If a matrix is both symmetric matrix and skew symmetric matrix then
(a) A is a diagonal matrix
(b) A is zero matrix
(c) A is scalar matrix
(d) None of these

Question 34.
If \(\left[\begin{array}{cc}
x+y & 3 \\
4 & x-y
\end{array}\right]\) = \(\left[\begin{array}{cc}
1 & 3 \\
4 & -3
\end{array}\right]\) then (x, y) is
(a) (-1, 2)
(b) (-1, -2)
(c) (-2, -1)
(d) (1, -2)

Question 35.
The matrix P = \(\left[\begin{array}{ccc}
0 & 0 & 4 \\
0 & 4 & 0 \\
4 & 0 & 0
\end{array}\right]\) is
(a) square matrix
(b) diagonal matrix
(c) unit matrix
(d) None of these

Question 36.
Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is
(a) 9
(b) 27
(c) 81
(d) 512

Question 37.
If \(\left[\begin{array}{cc}
2x+y & 4x \\
5x-7 & 4x
\end{array}\right]\) = \(\left[\begin{array}{cc}
7 & 7y-13 \\
y & x+6
\end{array}\right]\) then the value of x, y is
(a) 3, 1
(b) 2, 3
(c) 2, 4
(d) 3, 3

Question 38.
If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of matrix (5A – 2B) is
(a) m × 3
(b) 3 × 3
(c) m × n
(d) 3 × n

Question 39.
If A = \(\frac{1}{π}\) \(\left[\begin{array}{cc}
\sin ^{-1}(x \pi) & \tan^{1}\left(\frac{x}{\pi}\right) \\
\sin ^{-1}\left(\frac{x}{\pi}\right) & \cot ^{-1}(\pi x)
\end{array}\right]\)
B = \(\frac{1}{π}\) \(\left[\begin{array}{cc}
\cos ^{-1}(x \pi) & \tan ^{-1}\left(\frac{x}{\pi}\right) \\
\sin ^{-1}\left(\frac{x}{\pi}\right) & -\tan ^{-1}(\pi x)
\end{array}\right]\)
then A – B equal to
(a) I
(b) O
(c) 1
(d) \(\frac{3}{2}\) I

Question 40.
If A = \(\left[\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right]\) then A² is equal to
(a) \(\left[\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
1 & 0 \\
1 & 0
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
0 & 1 \\
0 & 1
\end{array}\right]\)
(d) \(\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]\)

Question 41.
If matrix A = [a_ij]_2×2 where a_ij = {\(_{0 if i = j}^{1 if i ≠ j}\) then A² is equal to
(a) I
(b) A
(c) O
(d) None of these

Question 42.
The matrix \(\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 0
\end{array}\right]\) is a
(a) identity matrix
(b) symmetric matrix
(c) skew symmetric matrix
(d) None of these

Question 43.
The matrix \(\left[\begin{array}{ccc}
0 & -5 & 8 \\
5 & 0 & 12 \\
-8 & -12 & 0
\end{array}\right]\) is a
(a) diagonal matrix
(b) symmetric matrix
(c) skew symmetric matrix
(d) scalar matrix

Question 44.
If A is matrix of order m × n and B is a matrix such that AB’ and B’A are both defined, then order of matrix B is
(a) m × m
(b) n × n
(c) n × m
(d) m × n

Question 45.
If A and B are matrices of same order, then (AB’ – BA’) is a
(a) skew symmetric matrix
(b) null matrix
(c) symmetric matrix
(d) unit matrix

Question 46.
If A is a square matrix such that A² = I, then (A – I)³ + (A + I)³ – 7A is equal to
(a) A
(b) I – A
(c) I + A
(d) 3 A

Question 47.
For any two matrices A and B, we have
(a) AB = BA
(b) AB ≠ BA
(c) AB = 0
(d) None of these

Question 48.
If A = [a_ij]_2×2 where a_ij = i + j, then A is equal to
(a) \(\left[\begin{array}{cc}
1 & 2 \\
3 & 4
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
2 & 3 \\
3 & 4
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
1 & 1 \\
2 & 2
\end{array}\right]\)
(d) \(\left[\begin{array}{cc}
1 & 2 \\
1 & 2
\end{array}\right]\)

Question 49.
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
(a) 18
(b) 512
(c) 81
(d) None of these

Question 50.
The order of the single matrix obtained from
\(\left[\begin{array}{cc}
1 & -1 \\
0 & 2 \\
2 & 3
\end{array}\right]\) \(\left\{\left[\begin{array}{ccc}
-1 & 0 & 2 \\
2 & 0 & 1
\end{array}\right]-\left[\begin{array}{ccc}
0 & 1 & 23 \\
1 & 0 & 21
\end{array}\right]\right\}\) is
(a) 2 × 2
(b) 2 × 3
(c) 3 × 2
(d) 3 × 3

Question 51.
A square matrix A = [a_ij]_n×n is called a diagonal matrix if a_ij = 0 for
(a) i = j
(b) i < j
(c) i > j
(d) i ≠ j

Question 52.
A square matrix A = [a_ij]_n×n is called a lower triangular matrix if a_ij = 0 for
(a) i = j
(b) i < j
(c) i > j
(d) None of these

Question 53.
The matrix A = \(\left[\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right]\) is a
(a) unit matrix
(b) diagonal matrix
(c) symmetric matrix
(d) skew symmetric matrix

Question 54.
If \(\left[\begin{array}{cc}
x+y & 2x+z\\
x-y & 2z+2
\end{array}\right]\) = \(\left[\begin{array}{cc}
4 & 7 \\
0 & 10
\end{array}\right]\) then find the value of x, y, z and w respectively
(a) 2, 2, 3, 4
(b) 2, 3, 1, 2
(c) 3, 3, 0, 1
(d) None of these

Question 55.
If \(\left[\begin{array}{cc}
x-y & 2x+z\\
2x-y & 3z+w
\end{array}\right]\) = \(\left[\begin{array}{cc}
-1 & 5 \\
0 & 13
\end{array}\right]\) then the value of w is
(a) 1
(b) 2
(c) 3
(d) 4

Question 56.
Find x, y, z and w respectively such that
\(\left[\begin{array}{cc}
x-y & 2x+z\\
2x-y & 2x+w
\end{array}\right]\) = \(\left[\begin{array}{cc}
5 & 3 \\
12 & 15
\end{array}\right]\)
(a) 7, 2, 1, 1
(b) 7, 5, 3, 8
(c) 1, 2, 5, 6
(d) 6, 3, 2, 1

Question 57.
If \(\left[\begin{array}{cc}
a+b & 2\\
5 & ab
\end{array}\right]\) = \(\left[\begin{array}{cc}
6 & 2 \\
5 & 8
\end{array}\right]\) then find the value of a and b respectively
(a) 2, 4
(b) 4, 2
(c) Both (a) and (b)
(d) None of these

Question 58.
For what values of x and y are the following matrices equal
A = \(\left[\begin{array}{cc}
2x+1 & 3y\\
0 & y^{2}-5y
\end{array}\right]\) B = \(\left[\begin{array}{cc}
x+3 & y^{2}+2 \\
0 & -6
\end{array}\right]\)
(a) 2, 3
(b) 3, 4
(c) 2, 2
(d) 3, 3

Question 59.
If A = \(\left[\begin{array}{cc}
α & 0\\
1 & 1
\end{array}\right]\) and B = \(\left[\begin{array}{cc}
1 & 0 \\
5 & 1
\end{array}\right]\) then find value of α for which A² = B is
(a) 1
(b) -1
(c) 4
(d) None of these

Question 60.
If P = \(\left[\begin{array}{ccc}
i & 0 & -i \\
0 & -i & i \\
-i & i & 0
\end{array}\right]\) and Q = \(\left[\begin{array}{cc}
-i & i \\
0 & 0 \\
i & -i
\end{array}\right]\) then PQ is equal to
(a) \(\left[\begin{array}{cc}
-2 & 2 \\
1 & -1 \\
1 & -1
\end{array}\right]\)
(b) \(\left[\begin{array}{cc}
2 & -2 \\
-1 & 1 \\
-1 & 1
\end{array}\right]\)
(c) \(\left[\begin{array}{cc}
2 & -2\\
-1 & 1
\end{array}\right]\)
(d) \(\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)

Question 61.
\(\left[\begin{array}{c}
1 & x & 1
\end{array}\right]\) \(\left[\begin{array}{ccc}
1 & 3 & 2 \\
2 & 5 & 1 \\
15 & 3 & 2
\end{array}\right]\) \(\left[\begin{array}{c}
1 \\
2 \\
x
\end{array}\right]\)
(a) -7
(b) -11
(c) -2
(d) 14

Question 62.
If A = \(\left[\begin{array}{cc}
1 & -1\\
2 & -1
\end{array}\right]\) B = \(\left[\begin{array}{cc}
x & 1\\
y & -1
\end{array}\right]\) and (A + B)² = A² + B², then x + y is
(a) 2
(b) 3
(c) 4
(d) 5

Question 63.
If AB = A and BA = B, then
(a) B = 1
(b)A = I
(c) A² = A
(d) B² = I

Question 64.
If A = \(\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
a & b & -1
\end{array}\right]\) then (A – I) (A + I) = 0 for
(a) a = b = 0 only
(b) a = 0 only
(c) b = 0 only
(d) any a and b

Question 65.
If A = \(\left[\begin{array}{cc}
1 & 1\\
0 & 2
\end{array}\right]\) then A⁸ – 2⁸ (A – I)
(a) I – A
(b) 2I – A
(c) I + A
(d) A – 2I

Question 66.
If A = \(\left[\begin{array}{ccc}
2 & 2 & 1 \\
1 & 3 & 1 \\
1 & 2 & 2
\end{array}\right]\) then A³ – 7A² + 10A =
(a) 5I + A
(b) 5I – A
(c) 5I
(d) 6I

Question 67.
If A is a m × n matrix such that AB and BA are both defined, then B is an
(a) m × n matrix
(b) n × m matrix
(c) n × n matrix
(d) m × m matrix

Question 68.
If A = \(\left[\begin{array}{cc}
1 & 2\\
3 & 4
\end{array}\right]\) then A² – 5A is equal to
(a) 2I
(b) 3I
(c) -2I
(d) null matrix

Question 69.
If A = \(\left[\begin{array}{cc}
-2 & 4\\
-1 & 2
\end{array}\right]\) then A² is
(a) null matrix
(b) unit matrix
(c) \(\left[\begin{array}{cc}
0 & 0\\
0 & 0
\end{array}\right]\)
(d) \(\left[\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right]\)

Question 70.
If A and B are 2 × 2 matrices, then which of the following is true?
(a) (A + B)² = A² + B² + 2AB
(b) (A – B)² = A² + B² – 2AB
(c) (A – B)(A + B) = A² + AB – BA – B²
(d) (A + B) (A – B) = A² – B²