Chapter 3 — Matrices

Class 12 Maths Chapter 3 covers Matrices — an ordered rectangular array of numbers or functions — including types of matrices, matrix operations (addition, scalar multiplication, matrix multiplication), transpose, symmetric and skew-symmetric matrices, and invertible matrices.

Chapter 3 of NCERT Class 12 Mathematics Part I introduces matrices as ordered rectangular arrays of numbers or functions. It covers the order of a matrix (m × n), types such as row, column, square, diagonal, scalar, identity, and zero matrices, and conditions for equality of matrices. The chapter explains operations: addition (same-order matrices only), scalar multiplication, and matrix multiplication (number of columns of A must equal number of rows of B). Key properties include non-commutativity of matrix multiplication, transpose properties, symmetric and skew-symmetric matrices, and invertible matrices with unique inverses.

1. 01A matrix of order m × n has m rows and n columns and mn elements total; element aij lies in the ith row and jth column.
2. 02Matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B; the product C = AB has order m × p if A is m × n and B is n × p.
3. 03Matrix multiplication is not commutative in general: AB ≠ BA even when both products are defined.
4. 04The transpose A′ of an m × n matrix is the n × m matrix with rows and columns interchanged; key property: (AB)′ = B′A′.
5. 05A square matrix A is symmetric if A′ = A, and skew-symmetric if A′ = −A; every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.
6. 06A square matrix A is invertible if there exists a matrix B such that AB = BA = I; the inverse A⁻¹ is unique, and (AB)⁻¹ = B⁻¹A⁻¹.