Answer:
[tex]A^{-1}=\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}[/tex]
Step-by-step explanation:
[tex]A=\begin{bmatrix}-7 & -2\\ 4 & 1\end{bmatrix}[/tex]
[tex]det\ A=-7\times 1-(-2)\times 4\\\Rightarrow det\ A=1[/tex]
[tex]A^{-1}=\frac{1}{det\ A}\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}\\\Rightarrow A^{-1}=\frac{1}{1}\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}\\\Rightarrow A^{-1}=\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}[/tex]
[tex]\therefore A^{-1}=\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}[/tex]
[tex]A.A^{-1}=\begin{bmatrix}-7 & -2\\ 4 & 1\end{bmatrix}\times \begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}\\\Rightarrow A.A^{-1}=\begin{pmatrix}\left(-7\right)\cdot \:1+\left(-2\right)\left(-4\right)&\left(-7\right)\cdot \:2+\left(-2\right)\left(-7\right)\\ 4\cdot \:1+1\cdot \left(-4\right)&4\cdot \:2+1\cdot \left(-7\right)\end{pmatrix}\\\Rightarrow A.A^{-1}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}[/tex]
Hence, confirmed
