Invertible matrices

A square matrix A of order m is invertible if there is a square matrix B of the same order with AB = BA = I. B is then the inverse of A, written A⁻¹.

Example: for A = [[3, 5], [1, 2]] and B = [[2, −5], [−1, 3]], the entries of AB are 6 − 5 = 1, −15 + 15 = 0, 2 − 2 = 0 and −5 + 6 = 1, so AB = I; BA = I as well, so B = A⁻¹ and A = B⁻¹.

Only square matrices can have inverses: for AB and BA both to be defined and both equal to one identity matrix, A and B must be square of the same order.

The relation runs both ways: whenever B undoes A, A also undoes B, so (A⁻¹)⁻¹ = A.

The inverse, when it exists, is unique. If B and C are both inverses of A, then B = BI = B(AC) = (BA)C = IC = C.

(AB)⁻¹ = B⁻¹A⁻¹ for invertible A and B of the same order, reversing the order just as the transpose does.

Not every square matrix is invertible. If AB = O for some non-zero B, A has no inverse; [[0, −1], [0, 2]] is one such matrix. The next chapter finds inverses with determinants: for a 2 × 2 matrix [[a, b], [c, d]] with ad − bc ≠ 0, the inverse is (1/(ad − bc))[[d, −b], [−c, a]].