Definition:Invertible Matrix

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Let $\struct {R, +, \circ}$ be a ring with unity.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$.

Then $\mathbf A$ is invertible if and only if:

$\exists \mathbf B \in \struct {\map {\MM_R} n, +, \times}: \mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$

where $\mathbf I_n$ denotes the unit matrix of order $n$.

Such a $\mathbf B$ is the inverse of $\mathbf A$.

It is usually denoted $\mathbf A^{-1}$.

Non-Invertible Matrix

Let $\mathbf A$ have no inverse.

Then $\mathbf A$ is referred to as non-invertible.

Also known as

An invertible matrix is called by some authors a non-singular matrix or regular matrix.

Also see