Matrices – Class 12 Practice Worksheet

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Section A — Multiple Choice Questions

Choose the single best answer | Each question carries 1 mark

Q 1

MCQ

A matrix of order m × n has 18 elements. Which of the following is a possible order?

A 4 × 5

B 3 × 6

C 2 × 9

D Both B and C

✅ Solution

For an m × n matrix, number of elements = mn = 18.
Factor pairs of 18: (1,18), (2,9), (3,6), (6,3), (9,2), (18,1).
Both 3×6 and 2×9 are valid.
Answer: (D) Both B and C

Q 2

MCQ

If A = [a_ij]_m×n is a square matrix, then which condition must hold?

A m < n

B m > n

C m = n

D m ≠ n

✅ Solution

A square matrix has equal number of rows and columns.
Answer: (C) m = n

Q 3

MCQ

The element a₂₃ of a matrix A = [a_ij] where a_ij = ½|i − 3j| is:

A 1/2

B 2

C 3/2

D 7/2

✅ Solution

a₂₃ = ½|2 − 3×3| = ½|2 − 9| = ½ × 7 = 7/2
Answer: (D) 7/2

Q 4

MCQ

If a matrix has 24 elements, which of the following is NOT a possible order?

A 4 × 6

B 3 × 8

C 5 × 5

D 2 × 12

✅ Solution

5 × 5 = 25 ≠ 24. All others are factor pairs of 24.
Answer: (C) 5 × 5

Q 5

MCQ

If A = [a_ij]_3×3 is a diagonal matrix with a₁₁=2, a₂₂=−1, a₃₃=5, what is a₁₂?

A 2

B −1

C 5

D 0

✅ Solution

In a diagonal matrix, all non-diagonal elements are 0. Since 1 ≠ 2, a₁₂ = 0.
Answer: (D) 0

Q 6

MCQ

If A = [[2,4],[3,2]] and B = [[1,3],[−2,5]], what is the element at position (1,2) in A + B?

A 5

B 7

C 8

D 3

✅ Solution

(A+B)₁₂ = a₁₂ + b₁₂ = 4 + 3 = 7
Answer: (B) 7

Q 7

MCQ

For two matrices A (m×n) and B (n×p), the order of the product AB is:

A n × n

B m × p

C m × n

D p × m

✅ Solution

If A is m×n and B is n×p, then AB is m×p. The inner dimensions must match.
Answer: (B) m × p

Q 8

MCQ

If A = [[3,−2],[4,−2]] and I = [[1,0],[0,1]], find k so that A² = kA − 2I.

A k = 1

B k = 2

C k = 3

D k = 4

✅ Solution

A² = [[3,−2],[4,−2]]×[[3,−2],[4,−2]] = [[9−8, −6+4],[12−8, −8+4]] = [[1,−2],[4,−4]]
kA − 2I = [[3k−2, −2k],[4k, −2k−2]]
Comparing: 3k−2=1 → k=1; −2k=−2 → k=1. ✓
Answer: (A) k = 1

Q 9

MCQ

A square matrix A is skew symmetric if:

A A' = A

B A' = −A

C A = I

D A' = A⁻¹

✅ Solution

By definition, A is skew symmetric if A' = −A. Diagonal elements must all be 0.
Answer: (B) A' = −A

Q 10

MCQ

The number of all possible matrices of order 3×3 with each entry 0 or 1 is:

A 27

B 18

C 81

D 512

✅ Solution

A 3×3 matrix has 9 entries. Each can be 0 or 1 (2 choices).
Total = 2⁹ = 512
Answer: (D) 512

Q 11

MCQ

If A and B are symmetric matrices of same order, then AB − BA is a:

A Symmetric matrix

B Skew symmetric matrix

C Zero matrix

D Identity matrix

✅ Solution

(AB−BA)' = (AB)' − (BA)' = B'A' − A'B' = BA − AB = −(AB−BA)
So AB−BA is skew symmetric.
Answer: (B) Skew symmetric matrix

Q 12

MCQ

If A = [[2,3],[1,2]] and B = [[2,−3],[−1,2]], then AB = ?

A [[1,0],[0,1]]

B [[0,0],[0,0]]

C [[4,6],[2,4]]

D [[3,0],[0,3]]

✅ Solution

AB = [[4−3, −6+6],[2−2, −3+4]] = [[1,0],[0,1]] = I
This means B = A⁻¹.
Answer: (A) [[1,0],[0,1]]

Q 13

MCQ

The transpose of a matrix A = [[1,2,3],[4,5,6]] has order:

A 2 × 3

B 3 × 2

C 1 × 6

D 6 × 1

✅ Solution

A is 2×3. Transpose A' swaps rows and columns, so A' is 3×2.
Answer: (B) 3 × 2

Q 14

MCQ

Matrices A and B are inverses of each other only if:

A AB = BA

B AB = BA = O

C AB = O, BA = I

D AB = BA = I

✅ Solution

By definition of inverse matrix: B = A⁻¹ if and only if AB = BA = I.
Answer: (D) AB = BA = I

Q 15

MCQ

If A is a square matrix such that A² = A, then (I + A)³ − 7A equals:

A A

B I − A

C I

D 3A

✅ Solution

(I+A)³ = I³ + 3I²A + 3IA² + A³ = I + 3A + 3A + A (since A²=A, A³=A)
= I + 7A
So (I+A)³ − 7A = I + 7A − 7A = I
Answer: (C) I

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Section B — Short Answer (Numerical)

Write the answer in one word / number | Each question carries 2 marks

Q 16

Short

If a_ij = (i + 2j)² / 2, find the element a₁₁ of the 2×2 matrix.

✅ Solution

a₁₁ = (1 + 2×1)² / 2 = (3)² / 2 = 9/2
Answer: 9/2

Q 17

Short

Find x if [[4,3],[x,5]] = [[y,z],[1,5]] — specifically, what is the value of x?

✅ Solution

By equality of matrices, corresponding elements are equal.
x = 1 (comparing (2,1) elements)
Answer: x = 1

Q 18

Short

If 2X + Y = [[1,0],[3,2]] and X + 2Y = [[−1,6],[0,2]], find the sum of all elements of matrix X.

✅ Solution

Subtracting 2nd from 2×(1st): 3X = 2[[1,0],[3,2]] − [[−1,6],[0,2]] = [[3,−6],[6,2]]
X = [[1, −2],[2, 2/3]]
Sum = 1 + (−2) + 2 + 2/3 = 1 + 2/3 = 5/3
Alternative: Use eqn (1)×2 − eqn (2): 3X = [[3,−6],[6,2]] → X = [[1,−2],[2,2/3]] Answer: 5/3

Q 19

Short

If A = [[1,2],[3,4]], calculate the element at position (1,1) of A².

✅ Solution

A² = A×A. The (1,1) element = (row 1 of A) · (col 1 of A) = 1×1 + 2×3 = 1 + 6 = 7
Answer: 7

Q 20

Short

For matrix A = [[1,5],[6,7]], find the (2,1) element of (A − A').

✅ Solution

A' = [[1,6],[5,7]]
A − A' = [[0,−1],[1,0]]
(A−A')₂₁ = 1. Note: A−A' is always skew-symmetric.
Answer: 1

Q 21

Short

If A = [[cosθ, sinθ],[−sinθ, cosθ]], find the value of A² at θ = π/2 — specifically element (1,2).

✅ Solution

Using A^n = [[cos(nθ), sin(nθ)],[−sin(nθ), cos(nθ)]], for n=2 and θ=π/2:
A² = [[cos π, sin π],[−sin π, cos π]] = [[−1,0],[0,−1]]
Element (1,2) = 0
Answer: 0

Q 22

Short

Find the value of y if [[x+y, 2],[5+z, xy]] = [[6,2],[5,8]].

✅ Solution

From (1,1): x + y = 6
From (2,2): xy = 8
So x and y are roots of t² − 6t + 8 = 0 → (t−2)(t−4) = 0
x = 2, y = 4 or x = 4, y = 2
Answer: y = 2 or y = 4

Q 23

Short

How many elements does a matrix of order 4 × 3 have?

✅ Solution

Number of elements = m × n = 4 × 3 = 12
Answer: 12

Q 24

Short