# Matrix inverse

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

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

where In 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.