Matrix inverse

In matrix algebra, the inverse of a square matrix A is X if

$\mathbf {AX} =\mathbf {XA} =\mathbf {I} _{n}\$

where I_n is the n-by-n identity matrix

If this equation is true, X is the inverse of A, denoted by A^-1. A is also the inverse of X.

A matrix is invertible if and only if it possesses an inverse.

Uniqueness

Every invertible matrix has only one inverse.

For example, if AX = I and AY = I, then X = Y. So, X = Y = A^-1.

To prove this, consider the case of X.A.Y.

Calculation

The inverse may be computed from the adjugate matrix, which shows that a matrix is invertible if and only if its determinant is itself invertible: over a field such as the real or complex numbers, this is equivalent to specifying that the determinant does not equal zero.