Matrix Inverses

Why

What is the inverse element under matrix multiplication.

Definition

Recall that if $A \in \R ^{m \times n}$ then $x \mapsto Ax$ is a function from $\R ^{n}$ to $\R ^{m}$. Clearly, if $m \neq n$, then the inverse of $f$ can not exist.¹

Now suppose that $A \in \R ^{n \times n}$. Of course, the inverse may not exist. Consider, for example if $A$ was the $n$ by $n$ matrix of zeros. If there exists a matrix $B$ so that $BA = I$ we call $B$ the left inverse of $A$ and likewise if $AC = I$ we call $C$ the right inverse of $A$. In the case that $A$ is square, the right inverse and left inverse coincide.

Let $A, B, C \in \R ^{n \times n}$. Let $BA = I$ and $AC = I$. Then $B = C$.

Since $BA = AC$ we have $BBA = BAC$ so $B = C$ since $BA = I$.

Notation

Let $\F $ be a field. Let $A \in \F ^{n \times n}$ be invertible. We follow the notation of inverse elements and denote the inverse of $A$ by $A^{-1}$.

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