Inverse of a Matrix

Definition. If $A$ is a square matrix of order $m$ and there exists another square matrix $B$ of the same order such that $AB = BA = I$, where $I$ is the identity matrix of order $m$, then $B$ is called the inverse of $A$, written $A^{-1}$. Writing $A^{-1}$ in place of $B$, the defining relation becomes $AA^{-1} = A^{-1}A = I$; by the very same definition, $A$ is then equally the inverse of $B$, so $B^{-1} = A$.

Worked illustration. If $A = \begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $B = \begin{bmatrix}2&-3\\-1&2\end{bmatrix}$, then $AB = \begin{bmatrix}2&3\\1&2\end{bmatrix}\begin{bmatrix}2&-3\\-1&2\end{bmatrix} = \begin{bmatrix}4-3&-6+6\\2-2&-3+4\end{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix} = I_2$. Likewise $BA = \begin{bmatrix}2&-3\\-1&2\end{bmatrix}\begin{bmatrix}2&3\\1&2\end{bmatrix} = \begin{bmatrix}4-3&6-6\\-2+2&-3+4\end{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix} = I_2$. Since both products equal the identity, $B = A^{-1}$ and $A = B^{-1}$.

When does an inverse exist? Consider $A = \begin{bmatrix}1&2\\2&4\end{bmatrix}$: no matrix $X$ can be found with $AX = I$. The reason is that $|A| = 1\cdot4 - 2\cdot2 = 0$. This illustrates the necessary condition for a matrix $X$ with $AX=I$ to exist: $|A| \neq 0$, i.e. $A$ must be a non-singular matrix.

Three standing notes:

1. Every square matrix $A$ of order $m\times m$ has a corresponding determinant, $\det A = |A|$.
2. A matrix is said to be invertible if its inverse exists.
3. A square matrix $A$ has an inverse if and only if $|A| \neq 0$.

Uniqueness of the inverse. It can be proved that if $A$ is a square matrix with $|A|\neq0$, its inverse $A^{-1}$ is unique -- there is never more than one matrix that undoes $A$.

Theorem. If a square matrix $A$'s inverse exists, it is unique.

Proof. Let $A$ be a square matrix of order $m$ whose inverse exists. Suppose, if possible, that $B$ and $C$ are both inverses of $A$. By the definition of inverse, $AB=BA=I$ and $AC=CA=I$. Now consider

Hence $B=C$, i.e. the inverse -- when it exists -- is unique. …