The Inverse of a Matrix $\mathbf{A}$ , Denoted By $\mathbf{A}^{-1}$ , Is Such That If

The inverse of a matrix $\mathbf{A}$ , denoted by $\mathbf{A}^{-1}$ , is such that if $\vec{v}=\mathbf{A} \vec{u}$ , then $\vec{u}=\mathbf{A}^{-1} \vec{v}$ . For a matrix $\mathbf{A}=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ , the inverse $\mathbf{A}^{-1}$ is given by $\mathbf{A}^{-1}=\frac{1}{D}\left(\begin{array}{cc}d & -b \\ -c & a\end{array}\right)$ , where $D=a d-b c \mathbf{A}^{-1}$ is undefined if $D=0 .$ Let $\mathbf{A}=\left(\begin{array}{cc}1 & 1 \\ 2 & -2\end{array}\right)$ . Does $\mathbf{A}^{-1}=\frac{1}{-4}\left(\begin{array}{cc}-2 & -2 \\ -1 & 1\end{array}\right) ?$