# Simple matrix question about $\begin{bmatrix}a & b\\c & d\end{bmatrix}$

I have a question about this matrix: $\begin{bmatrix}a & b\\c & d\end{bmatrix}$

Show that if $D = ad − cb$ does not equal $0$, then $A^{-1}$ = $D^{-1}*\begin{bmatrix}d & -b\\-c & a\end{bmatrix}$

How do I show this? Can someone help?

• Hint: multiply the first matrix by the second. – Gribouillis Dec 14 '17 at 13:40

By definition, the inverse matrix $A^{-1}$ satisfies $$A A^{-1} = A^{-1} A = I$$ where $I$ is the (in this case, $2\times 2$) identity matrix. To show the matrix $B = \frac{1}{D}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ is equal to $A^{-1}$, it suffices to show $AB = I$.