Matrices

part 1

Basics of Matrices

• Concept and notation of matrices
• Order and equality of matrices
• Types of matrices — zero, identity, diagonal, scalar
• Addition of matrices and scalar multiplication

part 2

Matrix Multiplication

• Multiplication of matrices
• Non-commutativity of matrix multiplication
• Transpose of a matrix
• Symmetric and skew-symmetric matrices

part 3

Elementary Operations and Invertible Matrices

• Elementary row and column operations
• Invertible matrices
• Finding inverse using elementary operations

(AB)ᵀ = BᵀAᵀ

$(AB)^T = B^T A^T$

★ board favourite

Inverse of A

$A^{-1} = \frac{\text{adj}(A)}{|A|}$ (|A| ≠ 0)

★ board favourite

(Aᵀ)ᵀ = A

$(A^T)^T = A$

★ frequently asked

(A+B)ᵀ = Aᵀ + Bᵀ

$(A+B)^T = A^T + B^T$

★ frequently asked

Symmetric matrix

$A^T = A$

★ frequently asked

Skew-symmetric matrix

$A^T = -A$ (diagonal elements = 0)

★ frequently asked

Matrix multiplication

$(AB)_{ij} = \sum_k a_{ik} \cdot b_{kj}$

★ frequently asked

(AB)⁻¹ = B⁻¹A⁻¹

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

★ frequently asked

AA⁻¹ = A⁻¹A = I

$AA^{-1} = A^{-1}A = I$

★ frequently asked

Find the matrix X such that 2A + B + X = O, where A = [[−1, 2],[3, 4]] and B = [[3, −2],[1, 5]].

If A and B are symmetric matrices of the same order, prove that AB − BA is a skew-symmetric matrix.

If a matrix has 5 elements, write all possible orders it can have.

If A = [[2, 0, 1],[2, 1, 3],[1, −1, 0]], find A² − 5A + 4I and hence find a matrix X such that A² − 5A + 4I + X = O.

Express the matrix A = [[3, 5], [1, −1]] as the sum of a symmetric and a skew-symmetric matrix.

Construct a 2×2 matrix A = [aᵢⱼ] whose elements are given by aᵢⱼ = (i + j)²/2.

Find the values of x and y if 2[[1, 3],[0, x]] + [[y, 0],[1, 2]] = [[5, 6],[1, 8]].

If A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]], show that A² − 4A − 5I = O, where I is the 3×3 identity matrix.

Express the matrix A = [[1, 3, 5],[−6, 8, 3],[−4, 6, 5]] as the sum of a symmetric and a skew-symmetric matrix.

How many marks is Matrices worth in the Class 12 board exam?

Around 5 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Matrix multiplication and finding inverses by row operations.

Is Matrices easy or hard?

It's rated medium — a moderate-difficulty chapter in Class 12 Maths. Most students need about 3 hours to cover it well.

What are the important topics in Matrices?

The chapter covers 10 NCERT topics in 3 areas: Basics of Matrices; Matrix Multiplication; Elementary Operations and Invertible Matrices.

What questions come from Matrices in board exams?

Between 2019–2024, CBSE board papers asked questions from this chapter on Matrix Equations, Transpose Properties, Order of a Matrix, Matrix Multiplication — 1- to 5-mark questions.