# Left and Right Inverses of Square Matrix over Field are Equal It has been suggested that this page or section be merged into Left or Right Inverse of Matrix is Inverse. (Discuss)

## Theorem

Let $\Bbb F$ be a field, usually one of the standard number fields $\Q$, $\R$ or $\C$.

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

Let $\map \MM n$ denote the matrix space of order $n$ square matrices over $\Bbb F$.

Let $\mathbf B$ be a left inverse matrix of $\mathbf A$.

Then $\mathbf B$ is also a right inverse matrix of $\mathbf A$.

Similarly, let $\mathbf B$ be a right inverse matrix of $\mathbf A$.

Then $\mathbf B$ is also a right inverse matrix of $\mathbf A$.

## Proof

Consider the algebraic structure $\struct {\map \MM {m, n}, +, \circ}$, where:

$+$ denotes matrix entrywise addition
$\circ$ denotes (conventional) matrix multiplication.

From Ring of Square Matrices over Field is Ring with Unity, $\struct {\map \MM {m, n}, +, \circ}$ is a ring with unity.

Hence a fortiori $\struct {\map \MM {m, n}, +, \circ}$ is a monoid.

The result follows directly from Left Inverse and Right Inverse is Inverse.

$\blacksquare$