### Inverse - If a matrix is square and invertible, there exists a unique inverse $A$ such that: matrices, $AA^{-1}=A^{-1}A=I$ - A tall matrix has a left inverse and a right inverse if it is wide. - The solution to the linear system $Ax=b$ is $x=A^{-1}b$ - $(AB)^{-1}=B^{-1}A^{-1}$ - $\begin{bmatrix}a & b \\ c & d\end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$ - To be invertible, a matrix must not be singular (gaussian elimination results in a row of zeros) - Let $A$ and $B$ be square matrices and suppose that $A$ is invertible. Then $B$ is invertible if and only if $AB$ is invertible. [[linear systems]] describes gaussian elimination and other methods for computing the inverse to do this Singular matrices: - Noninvertible - $Ax=b$ has no solution for some $b$