Equal matrices

Two matrices are equal when they have the same order and every entry of one equals the entry in the same position of the other: aᵢⱼ = bᵢⱼ for all i and j.

Same entries in a different arrangement do not count: a 2 × 2 matrix and its rows swapped are different matrices unless the rows happen to be equal.

Matrices of different orders are never equal, even if they contain the same numbers.

A matrix equation is a bundle of ordinary equations, one per position. Equating entries of a 3 × 3 equation gives 9 equations.

Worked example: matching two 3 × 3 matrices position by position gives six one-line equations (for example x + 3 = 0 and 2y − 7 = 3y − 2), and solving each gives x = −3, y = −5, z = 2, a = −2, b = −7 and c = −1.

Worked example: 2a + b = 4 and a − 2b = −3 give a = 1, b = 2; 5c − d = 11 and 4c + 3d = 24 give c = 3, d = 4.