# Matrix inverse

From Citizendium, the Citizens' Compendium

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

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.