CBSE Class 12 · Chapter 3 · Algebra · 2026-27

Matrices Class 12: MCQ and case study questions

15 multiple-choice questions and 2 case-based questions on the current (rationalised) syllabus, in the style of the board paper's Sections A and E. Try each one first; the answer, a one-line reason and a worked solution open on a tap.

15 MCQs (1 mark each)
2 case studies (4 marks each)
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Multiple-choice questions

Tap an option to check it; the answer and a worked solution open. Section A of the board paper has 18 MCQs of 1 mark each, spread over all chapters.

Q1

·1 mark·Multiple choiceOrder, types and equality of matrices

$A = [a_{i j}]$ is a $2 \times 3$ matrix with $a_{i j} = i + 2 j$ . Then $a_{23}$ is

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Answer: (b)

8

Why: Row i = 2, column j = 3: 2 + 2 × 3.

$a_{23} = 2 + 2 (3) = 8$ .

Q2

·1 mark·Multiple choiceTranspose, symmetric and skew-symmetric matrices

If $A = [\begin{matrix} 3 & - 2 \\ 1 & 5 \end{matrix}]$ , then $A + A^{T}$ is

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Answer: (c)

[\begin{matrix} 6 & - 1 \\ - 1 & 10 \end{matrix}]

Why: Aᵀ swaps rows and columns: Aᵀ = [[3, 1], [−2, 5]].

$A^{T} = [\begin{matrix} 3 & 1 \\ - 2 & 5 \end{matrix}]$ , so $A + A^{T} = [\begin{matrix} 6 & - 1 \\ - 1 & 10 \end{matrix}]$ (symmetric, as always).

Q3

·1 mark·Multiple choiceOperations on matrices

$A$ is a $4 \times 3$ matrix and $B$ is a $3 \times 2$ matrix. Which statement is true?

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Answer: (a)

A B

is of order

4 \times 2

and

B A

is not defined

Why: AB: (4 × 3)(3 × 2) = 4 × 2; BA: (3 × 2)(4 × 3) needs 2 = 4.

$A B$ : columns of $A$ ( $3$ ) $=$ rows of $B$ ( $3$ ), order $4 \times 2$ . $B A$ : columns of $B$ ( $2$ ) $\neq$ rows of $A$ ( $4$ ), so $B A$ is not defined.

Q4

·1 mark·Multiple choiceOrder, types and equality of matrices

If $[\begin{matrix} x + y & 2 \\ 5 & x - y \end{matrix}] = [\begin{matrix} 6 & 2 \\ 5 & 2 \end{matrix}]$ , then $x y$ equals

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Answer: (d)

8

Why: Equal matrices: x + y = 6 and x − y = 2, so x = 4, y = 2.

$x + y = 6$ , $x - y = 2 \Rightarrow x = 4$ , $y = 2$ ; $x y = 8$ .

Q5

·1 mark·Multiple choiceOperations on matrices

If $A = [\begin{matrix} 2 & 1 \\ 0 & 3 \end{matrix}]$ , then $A^{2}$ is

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Answer: (b)

[\begin{matrix} 4 & 5 \\ 0 & 9 \end{matrix}]

Why: Multiply rows by columns: the (1,2) entry is 2·1 + 1·3 = 5.

$A^{2} = [\begin{matrix} 2 & 1 \\ 0 & 3 \end{matrix}] [\begin{matrix} 2 & 1 \\ 0 & 3 \end{matrix}] = [\begin{matrix} 4 + 0 & 2 + 3 \\ 0 + 0 & 0 + 9 \end{matrix}] = [\begin{matrix} 4 & 5 \\ 0 & 9 \end{matrix}]$ . (Squaring each entry is wrong.)

Q6

·1 mark·Multiple choiceTranspose, symmetric and skew-symmetric matrices

The diagonal entries of a skew-symmetric matrix are

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Answer: (c) all zero

Why: a_ii = −a_ii forces a_ii = 0.

Skew-symmetric means $A^{T} = - A$ , i.e. $a_{j i} = - a_{i j}$ . For $i = j$ : $a_{i i} = - a_{i i}$ , so $a_{i i} = 0$ .

Q7

·1 mark·Multiple choiceTranspose, symmetric and skew-symmetric matrices

The matrix $[\begin{matrix} 3 & x \\ 5 & 2 \end{matrix}]$ is symmetric when $x$ equals

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Answer: (b)

5

Why: Symmetric means a₁₂ = a₂₁.

$A^{T} = A$ requires $a_{12} = a_{21}$ , i.e. $x = 5$ .

Q8

·1 mark·Multiple choiceTranspose, symmetric and skew-symmetric matrices

For any square matrix $A$ , the matrix $A + A^{T}$ is

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Answer: (a) symmetric

Why: (A + Aᵀ)ᵀ = Aᵀ + A.

$(A + A^{T})^{T} = A^{T} + (A^{T})^{T} = A^{T} + A$ , so $A + A^{T}$ equals its own transpose: symmetric.

Q9

·1 mark·Multiple choiceOperations on matrices

If $A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ and $B = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$ , then $A B$ is

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Answer: (a) the zero matrix

O

Why: Every row of A times every column of B gives 1 − 1 = 0.

$A B = [\begin{matrix} 1 - 1 & - 1 + 1 \\ 1 - 1 & - 1 + 1 \end{matrix}] = O$ , although $A \neq O$ and $B \neq O$ .

Q10

·1 mark·Multiple choiceOperations on matrices

If $A = [\begin{matrix} 4 & 1 \\ 2 & 3 \end{matrix}]$ and $A^{2} - k A + 10 I = O$ , then $k$ equals

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Answer: (c)

7

Why: A² = [[18, 7], [14, 11]]; compare the (1,2) entries: 7 − k = 0.

$A^{2} = [\begin{matrix} 18 & 7 \\ 14 & 11 \end{matrix}]$ . $A^{2} - k A + 10 I = [\begin{matrix} 28 - 4 k & 7 - k \\ 14 - 2 k & 21 - 3 k \end{matrix}] = O$ gives $k = 7$ .

Q11

·1 mark·Multiple choiceInvertible matrices

Which matrix is the inverse of $A = [\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}]$ ?

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Answer: (d)

[\begin{matrix} 1 & - 1 \\ - 1 & 2 \end{matrix}]

Why: Check AB = BA = I directly.

$[\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}] [\begin{matrix} 1 & - 1 \\ - 1 & 2 \end{matrix}] = [\begin{matrix} 2 - 1 & - 2 + 2 \\ 1 - 1 & - 1 + 2 \end{matrix}] = I$ , and the product in the other order is also $I$ .

Q12

·1 mark·Multiple choiceTranspose, symmetric and skew-symmetric matrices

For matrices $A$ and $B$ for which $A B$ is defined, $(A B)^{T}$ equals

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Answer: (b)

B^{T} A^{T}

Why: Transposing a product reverses the order.

$(A B)^{T} = B^{T} A^{T}$ (reversal law). $A^{T} B^{T}$ need not even be defined when $A$ and $B$ are not square.

Q13

·1 mark·Multiple choiceOrder, types and equality of matrices

The number of $2 \times 2$ diagonal matrices whose entries all belong to ${0, 1, 2}$ is

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Answer: (a)

9

Why: Off-diagonal entries must be 0; each of the 2 diagonal entries has 3 choices.

A diagonal matrix has $a_{12} = a_{21} = 0$ . The entries $a_{11}, a_{22}$ each have $3$ choices: $3^{2} = 9$ .

Q14

·1 mark·Multiple choiceOperations on matrices

If $A = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}]$ , then $A^{2}$ equals

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Answer: (b)

[\begin{matrix} 5 & 4 \\ 4 & 5 \end{matrix}]

Why: (1,1) entry 1 + 4 = 5; (1,2) entry 2 + 2 = 4.

$A^{2} = [\begin{matrix} 1 + 4 & 2 + 2 \\ 2 + 2 & 4 + 1 \end{matrix}] = [\begin{matrix} 5 & 4 \\ 4 & 5 \end{matrix}]$ .

Q15

·1 mark·Multiple choiceOperations on matrices

If $A = [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]$ , then $A^{3}$ is

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Answer: (c)

[\begin{matrix} 8 & 0 \\ 0 & 27 \end{matrix}]

Why: Powers of a diagonal matrix: raise each diagonal entry to the power.

For a diagonal matrix, $A^{2} = [\begin{matrix} 4 & 0 \\ 0 & 9 \end{matrix}]$ and $A^{3} = [\begin{matrix} 8 & 0 \\ 0 & 27 \end{matrix}]$ . ( $3 A$ would give $[\begin{matrix} 6 & 0 \\ 0 & 9 \end{matrix}]$ .)

Case study questions

Section E of the board paper has three case-based questions of 4 marks: a real-life passage, then parts of 1, 1 and 2 marks, with a choice (OR) on the 2-mark part. All Class 12 case study questions by chapter.

Case study 1: Canteen sales (4 marks)

A school canteen sells samosas, juice and sandwiches. The quantities sold on Monday and Tuesday are the rows of $Q = [\begin{matrix} 20 & 15 & 10 \\ 25 & 10 & 20 \end{matrix}]$ (columns: samosa, juice, sandwich). The prices in rupees form $P = [\begin{matrix} 10 \\ 20 \\ 30 \end{matrix}]$ .

(i) Write the order of

Q

and of

P

. [1 mark]

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Answer:

2 \times 3

and

3 \times 1

Q

2 \times 3

and

P

3 \times 1

(so

Q P

is defined, of order

2 \times 1

). A1

(ii) Find Monday's takings. [1 mark]

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Answer: ₹

800

20 (10) + 15 (20) + 10 (30) = 200 + 300 + 300 = ₹ 800

. A1

(iii) Find

Q P

and the total takings for the two days. [2 marks]

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Answer:

Q P = [\begin{matrix} 800 \\ 1050 \end{matrix}]

; ₹

1850

Q P = [\begin{matrix} 200 + 300 + 300 \\ 250 + 200 + 600 \end{matrix}] = [\begin{matrix} 800 \\ 1050 \end{matrix}]

M1; total

= ₹ 1850

. A1

OR On Wednesday the canteen sells exactly twice Monday's quantities. Use a scalar multiple of the first row of

Q

to find Wednesday's takings. [2 marks]

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Answer: ₹

1600

Wednesday's row

= 2 [\begin{matrix} 20 & 15 & 10 \end{matrix}] = [\begin{matrix} 40 & 30 & 20 \end{matrix}]

M1; takings

= 400 + 600 + 600 = ₹ 1600

. A1

Case study 2: Fertiliser mix (4 marks)

A farmer uses two fertilisers. One bag of $F_{1}$ contains $10$ kg of nitrogen and $4$ kg of phosphate; one bag of $F_{2}$ contains $5$ kg of nitrogen and $8$ kg of phosphate. This is recorded as $N = [\begin{matrix} 10 & 5 \\ 4 & 8 \end{matrix}]$ (rows: nitrogen, phosphate; columns: $F_{1}$ , $F_{2}$ ).

(i) If the farmer uses

4

bags of

F_{1}

and

6

bags of

F_{2}

, find the nutrients supplied using

N [\begin{matrix} 4 \\ 6 \end{matrix}]

. [1 mark]

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Answer:

70

kg nitrogen,

64

kg phosphate

N [\begin{matrix} 4 \\ 6 \end{matrix}] = [\begin{matrix} 40 + 30 \\ 16 + 48 \end{matrix}] = [\begin{matrix} 70 \\ 64 \end{matrix}]

. A1

(ii) Is

N

symmetric? Give a reason. [1 mark]

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Answer: No

No:

a_{12} = 5 \neq a_{21} = 4

, so

N^{T} \neq N

. A1

(iii) Show that

B = \frac{1}{60} [\begin{matrix} 8 & - 5 \\ - 4 & 10 \end{matrix}]

is the inverse of

N

. [2 marks]

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Answer:

N B = B N = I

N B = \frac{1}{60} [\begin{matrix} 80 - 20 & - 50 + 50 \\ 32 - 32 & - 20 + 80 \end{matrix}] = I

M1; similarly

B N = \frac{1}{60} [\begin{matrix} 60 & 0 \\ 0 & 60 \end{matrix}] = I

, so

B = N^{- 1}

. A1

OR Express

N

as the sum of a symmetric and a skew-symmetric matrix. [2 marks]

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Answer:

[\begin{matrix} 10 & \frac{9}{2} \\ \frac{9}{2} & 8 \end{matrix}] + [\begin{matrix} 0 & \frac{1}{2} \\ - \frac{1}{2} & 0 \end{matrix}]

\frac{1}{2} (N + N^{T}) = [\begin{matrix} 10 & \frac{9}{2} \\ \frac{9}{2} & 8 \end{matrix}]

(symmetric) M1 and

\frac{1}{2} (N - N^{T}) = [\begin{matrix} 0 & \frac{1}{2} \\ - \frac{1}{2} & 0 \end{matrix}]

(skew-symmetric); their sum is

N

. A1

Next steps for Matrices

This free set is separate from the chapter's question bank. On the Matrices chapter page the revision notes and the first step of the Route to 95 are free for everyone. CBSE Pro adds all 63 questions in the chapter bank (short and long answers, assertion–reason and more case studies) with their full step-marking schemes, the Route to 95 checkpoints with your progress saved, Skill Builders, the full common-mistakes library and AI marking of your written working.

Matrices: important questions What Pro includes

Original questions written by CBSE Math Revision for the CBSE 2026-27 syllabus; not taken from NCERT, NCERT Exemplar or CBSE papers. Every answer was checked twice, the second time by an independent reviewer. CBSE Math Revision is independent and not affiliated with CBSE or NCERT. Spotted a slip? Tell us and it goes in the corrections log.