Precalculus (6th Edition) Blitzer

If $A=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]$, the matrix $A$ is invertible if and only if $ad-bc\ne 0$.
Consider the matrix $A=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]$. Now, we will check if the matrix is invertible or not. The inverse of matrix $\left[ A \right]$ is equal to: ${{\left[ A \right]}^{-1}}=\frac{1}{ad-bc}\left[ \begin{matrix} d & -b \\ -c & a \\ \end{matrix} \right]$ Thus, $ad-bc\ne 0$ for the matrix to be invertible.