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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)
  • No calculator needed
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Multiple-choice questions

Choose one option. 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

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

  1. (a)\(7\)
  2. (b)\(8\)
  3. (c)\(5\)
  4. (d)\(6\)
Show answer
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 \(A = \begin{bmatrix}3 & -2\\ 1 & 5\end{bmatrix}\), then \(A + A^T\) is

  1. (a)\(\begin{bmatrix}6 & -4\\ 2 & 10\end{bmatrix}\)
  2. (b)\(\begin{bmatrix}0 & -3\\ 3 & 0\end{bmatrix}\)
  3. (c)\(\begin{bmatrix}6 & -1\\ -1 & 10\end{bmatrix}\)
  4. (d)\(\begin{bmatrix}6 & -1\\ -1 & 5\end{bmatrix}\)
Show answer
Answer: (c) \(\begin{bmatrix}6 & -1\\ -1 & 10\end{bmatrix}\)

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

\(A^T = \begin{bmatrix}3 & 1\\ -2 & 5\end{bmatrix}\), so \(A + A^T = \begin{bmatrix}6 & -1\\ -1 & 10\end{bmatrix}\) (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?

  1. (a)\(AB\) is of order \(4 \times 2\) and \(BA\) is not defined
  2. (b)\(AB\) is of order \(3 \times 3\)
  3. (c)Both \(AB\) and \(BA\) are defined
  4. (d)\(AB\) is of order \(2 \times 4\)
Show answer
Answer: (a) \(AB\) is of order \(4 \times 2\) and \(BA\) is not defined

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

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

Q4

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

If \(\begin{bmatrix}x + y & 2\\ 5 & x - y\end{bmatrix} = \begin{bmatrix}6 & 2\\ 5 & 2\end{bmatrix}\), then \(xy\) equals

  1. (a)\(6\)
  2. (b)\(4\)
  3. (c)\(12\)
  4. (d)\(8\)
Show answer
Answer: (d) \(8\)

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

\(x + y = 6\), \(x - y = 2 \Rightarrow x = 4\), \(y = 2\); \(xy = 8\).

Q5

·1 mark·Multiple choiceOperations on matrices

If \(A = \begin{bmatrix}2 & 1\\ 0 & 3\end{bmatrix}\), then \(A^2\) is

  1. (a)\(\begin{bmatrix}4 & 1\\ 0 & 9\end{bmatrix}\)
  2. (b)\(\begin{bmatrix}4 & 5\\ 0 & 9\end{bmatrix}\)
  3. (c)\(\begin{bmatrix}4 & 2\\ 0 & 6\end{bmatrix}\)
  4. (d)\(\begin{bmatrix}4 & 3\\ 0 & 9\end{bmatrix}\)
Show answer
Answer: (b) \(\begin{bmatrix}4 & 5\\ 0 & 9\end{bmatrix}\)

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

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

Q6

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

The diagonal entries of a skew-symmetric matrix are

  1. (a)any real numbers
  2. (b)all equal to \(1\)
  3. (c)all zero
  4. (d)equal to one another but non-zero
Show answer
Answer: (c) all zero

Why: a_ii = −a_ii forces a_ii = 0.

Skew-symmetric means \(A^T = -A\), i.e. \(a_{ji} = -a_{ij}\). For \(i = j\): \(a_{ii} = -a_{ii}\), so \(a_{ii} = 0\).

Q7

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

The matrix \(\begin{bmatrix}3 & x\\ 5 & 2\end{bmatrix}\) is symmetric when \(x\) equals

  1. (a)\(3\)
  2. (b)\(5\)
  3. (c)\(-5\)
  4. (d)\(2\)
Show answer
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

  1. (a)symmetric
  2. (b)skew-symmetric
  3. (c)always the zero matrix
  4. (d)always the identity matrix
Show answer
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 \(A = \begin{bmatrix}1 & 1\\ 1 & 1\end{bmatrix}\) and \(B = \begin{bmatrix}1 & -1\\ -1 & 1\end{bmatrix}\), then \(AB\) is

  1. (a)the zero matrix \(O\)
  2. (b)the identity matrix \(I\)
  3. (c)\(2A\)
  4. (d)\(B\)
Show answer
Answer: (a) the zero matrix \(O\)

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

\(AB = \begin{bmatrix}1 - 1 & -1 + 1\\ 1 - 1 & -1 + 1\end{bmatrix} = O\), although \(A \ne O\) and \(B \ne O\).

Q10

·1 mark·Multiple choiceOperations on matrices

If \(A = \begin{bmatrix}4 & 1\\ 2 & 3\end{bmatrix}\) and \(A^2 - kA + 10I = O\), then \(k\) equals

  1. (a)\(10\)
  2. (b)\(-7\)
  3. (c)\(7\)
  4. (d)\(3\)
Show answer
Answer: (c) \(7\)

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

\(A^2 = \begin{bmatrix}18 & 7\\ 14 & 11\end{bmatrix}\). \(A^2 - kA + 10I = \begin{bmatrix}28 - 4k & 7 - k\\ 14 - 2k & 21 - 3k\end{bmatrix} = O\) gives \(k = 7\).

Q11

·1 mark·Multiple choiceInvertible matrices

Which matrix is the inverse of \(A = \begin{bmatrix}2 & 1\\ 1 & 1\end{bmatrix}\)?

  1. (a)\(\begin{bmatrix}1 & 1\\ 1 & 2\end{bmatrix}\)
  2. (b)\(\begin{bmatrix}-1 & 1\\ 1 & -2\end{bmatrix}\)
  3. (c)\(\begin{bmatrix}2 & -1\\ -1 & 1\end{bmatrix}\)
  4. (d)\(\begin{bmatrix}1 & -1\\ -1 & 2\end{bmatrix}\)
Show answer
Answer: (d) \(\begin{bmatrix}1 & -1\\ -1 & 2\end{bmatrix}\)

Why: Check AB = BA = I directly.

\(\begin{bmatrix}2 & 1\\ 1 & 1\end{bmatrix}\begin{bmatrix}1 & -1\\ -1 & 2\end{bmatrix} = \begin{bmatrix}2 - 1 & -2 + 2\\ 1 - 1 & -1 + 2\end{bmatrix} = I\), 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 \(AB\) is defined, \((AB)^T\) equals

  1. (a)\(A^TB^T\)
  2. (b)\(B^TA^T\)
  3. (c)\(AB\)
  4. (d)\(BA\)
Show answer
Answer: (b) \(B^TA^T\)

Why: Transposing a product reverses the order.

\((AB)^T = B^TA^T\) (reversal law). \(A^TB^T\) 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

  1. (a)\(9\)
  2. (b)\(81\)
  3. (c)\(6\)
  4. (d)\(27\)
Show answer
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 \(A = \begin{bmatrix}1 & 2\\ 2 & 1\end{bmatrix}\), then \(A^2\) equals

  1. (a)\(\begin{bmatrix}1 & 4\\ 4 & 1\end{bmatrix}\)
  2. (b)\(\begin{bmatrix}5 & 4\\ 4 & 5\end{bmatrix}\)
  3. (c)\(\begin{bmatrix}2 & 4\\ 4 & 2\end{bmatrix}\)
  4. (d)\(\begin{bmatrix}5 & 2\\ 2 & 5\end{bmatrix}\)
Show answer
Answer: (b) \(\begin{bmatrix}5 & 4\\ 4 & 5\end{bmatrix}\)

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

\(A^2 = \begin{bmatrix}1 + 4 & 2 + 2\\ 2 + 2 & 4 + 1\end{bmatrix} = \begin{bmatrix}5 & 4\\ 4 & 5\end{bmatrix}\).

Q15

·1 mark·Multiple choiceOperations on matrices

If \(A = \begin{bmatrix}2 & 0\\ 0 & 3\end{bmatrix}\), then \(A^3\) is

  1. (a)\(\begin{bmatrix}6 & 0\\ 0 & 9\end{bmatrix}\)
  2. (b)\(\begin{bmatrix}8 & 27\\ 0 & 0\end{bmatrix}\)
  3. (c)\(\begin{bmatrix}8 & 0\\ 0 & 27\end{bmatrix}\)
  4. (d)\(\begin{bmatrix}5 & 0\\ 0 & 6\end{bmatrix}\)
Show answer
Answer: (c) \(\begin{bmatrix}8 & 0\\ 0 & 27\end{bmatrix}\)

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

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

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.

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 \(Q = \begin{bmatrix}20 & 15 & 10\\ 25 & 10 & 20\end{bmatrix}\) (columns: samosa, juice, sandwich). The prices in rupees form \(P = \begin{bmatrix}10\\ 20\\ 30\end{bmatrix}\).

(i) Write the order of \(Q\) and of \(P\). [1 mark]
Show answer
Answer: \(2 \times 3\) and \(3 \times 1\)
\(Q\) is \(2 \times 3\) and \(P\) is \(3 \times 1\) (so \(QP\) is defined, of order \(2 \times 1\)). A1
(ii) Find Monday's takings. [1 mark]
Show answer
Answer: ₹\(800\)
\(20(10) + 15(20) + 10(30) = 200 + 300 + 300 = \text{₹}800\). A1
(iii) Find \(QP\) and the total takings for the two days. [2 marks]
Show answer
Answer: \(QP = \begin{bmatrix}800\\ 1050\end{bmatrix}\); ₹\(1850\)
\(QP = \begin{bmatrix}200 + 300 + 300\\ 250 + 200 + 600\end{bmatrix} = \begin{bmatrix}800\\ 1050\end{bmatrix}\) M1; total \(= \text{₹}1850\). 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]
Show answer
Answer: ₹\(1600\)
Wednesday's row \(= 2\begin{bmatrix}20 & 15 & 10\end{bmatrix} = \begin{bmatrix}40 & 30 & 20\end{bmatrix}\) M1; takings \(= 400 + 600 + 600 = \text{₹}1600\). 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 \(N = \begin{bmatrix}10 & 5\\ 4 & 8\end{bmatrix}\) (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 \(N\begin{bmatrix}4\\ 6\end{bmatrix}\). [1 mark]
Show answer
Answer: \(70\) kg nitrogen, \(64\) kg phosphate
\(N\begin{bmatrix}4\\ 6\end{bmatrix} = \begin{bmatrix}40 + 30\\ 16 + 48\end{bmatrix} = \begin{bmatrix}70\\ 64\end{bmatrix}\). A1
(ii) Is \(N\) symmetric? Give a reason. [1 mark]
Show answer
Answer: No
No: \(a_{12} = 5 \ne a_{21} = 4\), so \(N^T \ne N\). A1
(iii) Show that \(B = \dfrac{1}{60}\begin{bmatrix}8 & -5\\ -4 & 10\end{bmatrix}\) is the inverse of \(N\). [2 marks]
Show answer
Answer: \(NB = BN = I\)
\(NB = \dfrac{1}{60}\begin{bmatrix}80 - 20 & -50 + 50\\ 32 - 32 & -20 + 80\end{bmatrix} = I\) M1; similarly \(BN = \dfrac{1}{60}\begin{bmatrix}60 & 0\\ 0 & 60\end{bmatrix} = I\), so \(B = N^{-1}\). A1
OR Express \(N\) as the sum of a symmetric and a skew-symmetric matrix. [2 marks]
Show answer
Answer: \(\begin{bmatrix}10 & \tfrac92\\ \tfrac92 & 8\end{bmatrix} + \begin{bmatrix}0 & \tfrac12\\ -\tfrac12 & 0\end{bmatrix}\)
\(\tfrac12(N + N^T) = \begin{bmatrix}10 & \tfrac92\\ \tfrac92 & 8\end{bmatrix}\) (symmetric) M1 and \(\tfrac12(N - N^T) = \begin{bmatrix}0 & \tfrac12\\ -\tfrac12 & 0\end{bmatrix}\) (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 Essentials adds all 40 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 and the full common-mistakes library. There is no AI marking on CBSE Math Revision: you check your work against the marking scheme.

Matrices: notes and questionsNCERT solutions, Chapter 3What Essentials 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 re-checked by computer algebra and 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.