Class 12 Maths NCERT Solutions Chapter 3 Matrices | Boundless Maths
Unit II · Algebra · Chapter 3 Matrices — Class 12 Maths NCERT Solutions

Class 12 Maths NCERT Solutions Chapter 3: Matrices

Free, step-by-step NCERT Solutions for all four exercises of this chapter, taking a rectangular array of numbers from its basic building blocks all the way to a matrix's own multiplicative "undo." You'll use the same handful of ideas — order, equality, the four core operations, the transpose — to classify a matrix by shape, prove whether it's symmetric or skew symmetric, split any square matrix into those two parts, and finally test whether it's invertible at all. Every solution here is worked the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year laid out right on this page.

5Exercises (incl. Misc.)
56Total Questions
2026-27CBSE Syllabus
100%Solved

Class 12 Maths NCERT Solutions Chapter 3 — Overview

This Class 12 Maths NCERT Solutions Chapter 3 hub covers Matrices, the chapter that introduces a rectangular array of numbers as a single mathematical object you can add, scale, multiply and transform according to its own set of rules. You'll learn to classify matrices by shape (row, column, square, diagonal, scalar, identity, zero), test when two matrices are equal, and carry out addition, scalar multiplication and matrix multiplication — the last of which behaves very differently from ordinary number multiplication, since order matters and AB is usually not equal to BA.

The chapter builds in a deliberate sequence: matrix basics and types first (Exercise 3.1), then the four core operations and their algebraic properties (Exercise 3.2), then transpose along with symmetric and skew symmetric matrices (Exercise 3.3), and finally invertible matrices (Exercise 3.4) — the idea that some square matrices have a multiplicative "undo," much like a reciprocal. Matrices also lays the groundwork directly needed for Chapter 4, Determinants, so getting comfortable here pays off immediately in the next chapter.

How the Chapter Builds

One Idea Leads to the Next

1

Matrix Basics & Types

Order, elements, and the seven standard types of matrices, plus equality of matrices. Exercise 3.1.

→
2

Operations on Matrices

Addition, scalar multiplication, and matrix multiplication, with all their algebraic properties. Exercise 3.2.

→
3

Transpose & Symmetric Matrices

Flipping rows and columns, and splitting any square matrix into symmetric and skew symmetric parts. Exercise 3.3.

→
4

Invertible Matrices

When AB = BA = I, B is the inverse of A — and that inverse, if it exists, is always unique. Exercise 3.4 (included on the Ex 3.3 page).

Quick Reference

Important Formulas — Chapter 3

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Algebra.

Matrix Basics & Types (§3.2–3.3)

General matrix & order

A = [a_{ij}]_{m \times n},\;\; 1 \le i \le m,\; 1 \le j \le n

A matrix with m rows and n columns has exactly mn elements.

Square, diagonal, scalar, identity

Square: m=n  ·  Diagonal: a_{ij}=0,\; i\neq j  ·  Scalar: diagonal entries all equal  ·  Identity: diagonal entries all 1

Every identity matrix is a scalar matrix, but not every scalar matrix is an identity matrix.

Equality of matrices

A = B \iff \text{same order and } a_{ij} = b_{ij}\;\; \forall\, i, j

Both conditions are required — same order alone is not enough.

Operations on Matrices (§3.4)

Addition & scalar multiplication

(A+B)_{ij} = a_{ij}+b_{ij},\quad (kA)_{ij} = k\,a_{ij}

Addition needs A and B to be of the same order; scalar multiplication never has an order restriction.

Matrix multiplication

(AB)_{ik} = \sum_{j=1}^{n} a_{ij}\,b_{jk}

Defined only when the number of columns of A equals the number of rows of B.

Key algebraic properties

(AB)C = A(BC),\;\; A(B+C)=AB+AC,\;\; AI=IA=A

Multiplication is associative and distributive over addition, but not commutative — AB ≠ BA in general.

Transpose, Symmetric & Skew Symmetric Matrices (§3.5–3.6)

Properties of transpose

(A')' = A,\;\; (kA)'=kA',\;\; (A+B)'=A'+B',\;\; (AB)'=B'A'

Note the reversal in the last rule — the product order flips when you take the transpose.

Symmetric & skew symmetric

A' = A \;\text{(symmetric)},\qquad A' = -A \;\text{(skew symmetric)}

Every diagonal entry of a skew symmetric matrix must be zero.

Any square matrix, split in two

A = \tfrac{1}{2}(A+A') + \tfrac{1}{2}(A-A')

The first part is always symmetric, the second always skew symmetric.

Invertible Matrices (§3.7)

Definition of the inverse

AB = BA = I \;\Rightarrow\; B = A^{-1}

Only a square matrix can be invertible — a rectangular matrix has no inverse.

Uniqueness & product rule

(AB)^{-1} = B^{-1}A^{-1}

The inverse of a square matrix, if it exists, is always unique — and note the reversed order here too.

Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all four exercises.

  • Adding or subtracting matrices of different order. A + B is only defined when A and B have exactly the same number of rows and the same number of columns — check this before writing a single entry.
  • Assuming AB = BA. Matrix multiplication is not commutative in general — even when both AB and BA are defined and of the same order, they can come out completely different.
  • Ignoring the column–row rule for multiplication. AB is defined only when the number of columns of A equals the number of rows of B — always check this first, since AB being defined doesn't guarantee BA is too.
  • Getting the transpose-of-a-product rule backwards. The correct identity is (AB)′ = B′A′, not A′B′ — the order reverses.
  • Assuming AB = O forces A = O or B = O. Unlike ordinary numbers, two non-zero matrices can multiply to give the zero matrix.
  • Forgetting that diagonal elements of a skew symmetric matrix must be zero. This follows directly from aii = −aii, and examiners frequently test it.
  • Mixing up the order of the product matrix. If A is m × n and B is n × p, then AB is m × p — not p × m, and not n × n.
  • Trying to invert a non-square matrix. Only square matrices can possibly have an inverse — check the shape before attempting AB = BA = I.
Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 3 — Choose an Exercise

3.1

Exercise 3.1

Order and elements of a matrix, constructing matrices from a formula, and equality of matrices · 10 questions

Solve Exercise 3.1 →
3.2

Exercise 3.2

Addition, scalar multiplication and multiplication of matrices, with proofs using their properties · 22 questions

Solve Exercise 3.2 →
3.3

Exercise 3.3 & 3.4

Transpose of a matrix, identifying/proving symmetric and skew symmetric matrices, plus the single Ex 3.4 question on invertible matrices · 13 questions

Solve Exercise 3.3 →
M

Miscellaneous Exercise

Mixed questions combining matrix algebra, symmetric/skew symmetric proofs, and induction-based problems · 11 questions

Solve Miscellaneous →

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Every formula for Algebra — matrices, determinants — in one printable PDF.

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Common Questions

Frequently Asked Questions

Quick answers about Chapter 3, Matrices.

How many exercises are there in Chapter 3, Matrices?
There are four main exercises — 3.1, 3.2, 3.3 and 3.4 — plus a Miscellaneous Exercise, totalling 56 questions across matrix types, matrix operations, transpose, symmetric/skew symmetric matrices, and invertible matrices.
Is this chapter important for the board exam?
Yes — Matrices and Determinants together form the Algebra unit, and questions on matrix operations, symmetric/skew symmetric proofs, and inverses appear almost every year, usually worth 4 to 6 marks combined.
Why is matrix multiplication not commutative?
Because AB and BA involve pairing rows and columns in a different order — even when both products are defined and of the same size, the sums that make up each entry generally come out different, so AB ≠ BA in general. Diagonal matrices of the same order are a rare exception where multiplication does commute.
What is the difference between a symmetric and a skew symmetric matrix?
A square matrix A is symmetric if A′ = A, meaning it's mirror-identical across its diagonal. It's skew symmetric if A′ = −A, which forces every diagonal element to be zero. Every square matrix can be split into a symmetric part, ½(A + A′), and a skew symmetric part, ½(A − A′).
What should I revise before starting this chapter?
Matrices don't lean on earlier calculus chapters, but you do need to be comfortable with basic algebraic manipulation and solving simultaneous linear equations, since several questions ask you to equate corresponding entries of two equal matrices and solve for unknowns.
Where can I find the official NCERT textbook for this chapter?
Matrices is Chapter 3 of the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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