Matrices — determinant and inverse

Basic operations on square matrices. The determinant of a $2\times2$ matrix:

$\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc$

for a $3\times3$ matrix it is found by Laplace (cofactor) expansion. The determinant decides invertibility: a matrix has an inverse only when $\det\neq0$. In that case:

$A^{-1}=\frac{1}{\det A}\cdot(\text{transpose of the cofactor matrix})$

The calculator returns the determinant and the full inverse matrix for the $2\times2$ and $3\times3$ dimensions.