An inverse of a square matrix A is a matrix X such that AX = I and XA = I. (A matrix that is not square cannot have an inverse, though it may have a left inverse or a right inverse.) A square matrix A may or may not have an inverse, but if it has then that inverse is unique and A is said to be invertible. A matrix is invertible if and only if it is not singular. Consequently, the term ‘non‐singular’ is sometimes used for ‘invertible’.
When det A ≠ 0, the matrix (1/detA) adjA is the inverse of A, where adjA is the adjugate of A. For example, the 2×2 matrix A below is invertible if ad − bc ≠ 0, and its inverse A−1 is as shown: