Matrix Inverse

Summary

• $A$ is a square $n×n$ matrix. We say that $A$ is invertible or nonsingular if there exists an $n×n$ matrix $B$ such that

$$AB=BA=I$$

• If no such matrix exist, we say that $A$ is noninvertible or singular .
• If $A$ is invertible, then the inverse of $A$ is denoted $A^{-1}$
• If

$$A=\begin{bmatrix}a & b\\c & d\end{bmatrix}$$

• then $A$ is invertible if and only if $ad-cb≠0$ . If $A$ is invertible then

$$A^{-1}= \frac{1}{ad-cb} \begin{bmatrix}d & -b\\-c & a\end{bmatrix}$$

• Results. $A,B$ are $n×n$ invertible matrices and $α≠0$ is a scalar.
• $A$ can have at most one inverse
• $(A^{-1})^{-1}=A$
• $(αA)^{-1}=α^{-1}A^{-1}$
• $(A^T)^{-1}=(A^{-1})^T$
• $AB$ is invertible and $(AB)^{-1}=B^{-1}A^{-1}$