Chapter 3 — Matrices

Chapter 3 of the Class 12 Maths NCERT textbook, "Matrices", covers the 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.

• Matrices as organised data — A matrix is introduced as a grid that stores numbers or functions by row and column, with its order (m × n) fixing its shape. Recognising special shapes — row, column, square, diagonal, scalar, identity and zero matrices — sets up the rules for working with them.
• Operating on matrices — The chapter defines when matrices can be added, scaled or multiplied, stressing that multiplication needs the inner dimensions to match and, unlike ordinary numbers, order matters: AB rarely equals BA even when both make sense.
• Structure and reversibility — Flipping rows and columns gives the transpose, which splits any square matrix into symmetric and skew-symmetric parts. The chapter closes with invertible matrices, whose unique inverse behaves like division and satisfies (AB)⁻¹ = B⁻¹A⁻¹.

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.