Matrix inversion

Matrix inversion computes the matrix inverse $A^{-1}$ of a square matrix satisfying $A{A}^{-1}=A^{-1}A=I$. It is possible exactly when $A$ has full rank. One method applies Gauss-Jordan elimination to $[A\mid I]$; another factors $A$ and solves a linear system for each column of $I$.

For $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with $ad-bc\ne 0$, the inverse is $(ad-bc{)}^{-1}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$. To solve a single numerical system $Ax=b$, a factorization followed by triangular solves is usually preferable to explicitly forming the inverse. A large condition number can amplify input errors regardless of the inversion method.