Matrix inverse

Given a $2 \times 2$ matrix $\mathbf A$ , its inverse $\mathbf A^{-1}$ is:

\begin{align*} \mathbf A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad \mathbf A^{-1} = \frac{1}{\det \mathbf A} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}. \end{align*}