Class 12 Mathematics Chapter 3 Matrices – Extra Questions with Answers

This set of questions on Matrices moves from basics like matrix order and the null matrix through to skew-symmetric matrices and the identity matrix.

Last Updated: September 23, 2026

How to Approach the HOTS Questions in This Chapter

Very Short Answer Questions (1 mark)

Q1. What is the order of a matrix with 3 rows and 2 columns?
Ans: 3×2.

Q2. What is a zero (null) matrix?
Ans: A matrix in which all elements are zero.

Q3. What is the condition for a skew-symmetric matrix?
Ans: A’=−A.

Q4. Can two matrices of different orders be added?
Ans: No, matrix addition is only defined for matrices of the same order.

Q5. What is the identity matrix of order 2?
Ans: [[1,0],[0,1]].

Short Answer Questions (2–3 marks)

Q6. If A=[[1,2],[3,4]] and B=[[5,6],[7,8]], find A+B.
Ans: A+B=[[1+5,2+6],[3+7,4+8]]=[[6,8],[10,12]].

Q7. If A=[[1,2],[3,4]], find A’ (transpose).
Ans: A’=[[1,3],[2,4]].

Q8. Determine if AB is defined for A of order 2×3 and B of order 3×4, and if so, state the order of AB.
Ans: Yes, defined since columns of A (3)=rows of B (3). Order of AB=2×4.

Higher-Order Thinking / Application Questions

Q9. Prove that every square matrix A can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix, and illustrate this decomposition using A=[[2,3],[1,4]].
Ans: Any square matrix A can be written as A=½(A+A’)+½(A−A’), where ½(A+A’) is always symmetric (since its transpose equals itself) and ½(A−A’) is always skew-symmetric (since its transpose equals its negative). For A=[[2,3],[1,4]], A’=[[2,1],[3,4]]. A+A’=[[4,4],[4,8]], so ½(A+A’)=[[2,2],[2,4]] (symmetric, verify: its transpose is [[2,2],[2,4]], same as itself). A−A’=[[0,2],[−2,0]], so ½(A−A’)=[[0,1],[−1,0]] (skew-symmetric, verify: its transpose is [[0,−1],[1,0]]=−[[0,1],[−1,0]]). Adding these: [[2,2],[2,4]]+[[0,1],[−1,0]]=[[2,3],[1,4]]=A, confirming the decomposition. This decomposition is unique because if A=P+Q=P′+Q′ for symmetric P,P′ and skew-symmetric Q,Q′, then P−P′=Q′−Q would have to be simultaneously symmetric and skew-symmetric, which forces it to be the zero matrix, proving P=P′ and Q=Q′.

Q10. Explain why matrix multiplication is not commutative in general (AB≠BA), even when both products are defined, using a concrete 2×2 example, and describe the practical implication this has for solving matrix equations.
Ans: Let A=[[1,1],[0,1]] and B=[[1,0],[1,1]], both 2×2 square matrices, so both AB and BA are defined and have the same order. Computing AB: row1 of A times columns of B gives [1(1)+1(1), 1(0)+1(1)]=[2,1]; row2 of A times columns of B gives [0(1)+1(1), 0(0)+1(1)]=[1,1]. So AB=[[2,1],[1,1]]. Computing BA: row1 of B times columns of A gives [1(1)+0(0), 1(1)+0(1)]=[1,1]; row2 of B times columns of A gives [1(1)+1(0), 1(1)+1(1)]=[1,2]. So BA=[[1,1],[1,2]]. Since AB=[[2,1],[1,1]]≠BA=[[1,1],[1,2]], this concretely demonstrates that matrix multiplication is not commutative. This has an important practical implication: when solving matrix equations, one must always carefully preserve the order in which matrices are multiplied on each side of an equation (e.g., if AX=B, we must multiply both sides by A⁻¹ specifically on the LEFT to get X=A⁻¹B, since multiplying on the right, XA⁻¹, would generally give a different, incorrect result) — unlike ordinary number arithmetic where multiplication order never matters, matrix algebra requires careful attention to left versus right multiplication at every step.

Frequently Asked Questions

How would you find the value of x if a 2×2 matrix with entries x, 3, 2, 4 has determinant 2?
Determinant = x times 4 minus 3 times 2 = 4x minus 6 = 2, giving x = 2.

If matrix A is a 3×3 matrix and A is invertible, what does this tell you about its determinant?
An invertible matrix must have a non zero determinant, since the inverse is defined as the adjoint divided by the determinant.

Written by Satish

NCERTBooks.org is an independent educational resource run by a small team focused on making official NCERT textbooks easy to find, read, and download for students, parents, and teachers across India. We are not affiliated with NCERT or the Ministry of Education -- we organise publicly available NCERT content by class and subject, verify links against official sources, and build tools (like our in-browser reader) that make studying more convenient. Every guide we publish is written and reviewed by our team based on the actual NCERT curriculum and syllabus.