## Formula for 3×3 Matrix Inverse

Let $\displaystyle A=\begin{pmatrix}a &b &c\\ d& e& f\\ g& h& i \end{pmatrix}.$

Then,
$\displaystyle A^{-1}=\frac{1}{\det(A)}\begin{pmatrix} ei-fh &ch-bi &bf-ce\\ fg-di &ai-cg &cd-af\\ dh-eg &bg-ah &ae-bd \end{pmatrix}.$

This method is about the same speed as the Gaussian-Elimination method. However, if you have already calculated det(A), using the formula may be slightly faster.

The formula provides an insight on why A is singular (not invertible) if det(A)=0. (Because, if det(A)=0, in the formula we would be dividing by 0 which is not allowed.)