Invertible matrix

Definition

Let $R$ be a commutative ring and let $A$ be an $n \times n$ square matrix over $R$. The matrix $A$ is invertible when there is an $n \times n$ matrix $B$ over $R$ such that $$AB = BA = I_n,$$ where $I_n$ is the identity matrix. The matrix $B$ is then unique; it is the inverse of $A$, written $A^{-1}$.

Remarks

Over a field, other names are nonsingular and nondegenerate. A matrix over $R$ is invertible if and only if its determinant is a unit of $R$. Over a field, a one-sided inverse is two-sided, and $A$ is invertible if and only if $Ax = 0$ has only the solution $x = 0$. The matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is not invertible over $\R$, because its determinant is $0$.

Explanations

L1 human

A matrix is invertible when another matrix undoes it. If $A$ sends $x$ to $y$, then $A^{-1}$ sends $y$ back to $x$. For $2 \times 2$ matrices, $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ when $ad - bc \ne 0$.