Question

Find the inverse of ????−7 −2

4 1
????. Show your work. Confirm that the matrices are inverses.

Answer 1

`A^(-1)=\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}`

Explanation:

`A=\begin{bmatrix}-7 & -2\\ 4 & 1\end{bmatrix}`

`det\ A=-7* 1-(-2)* 4\\\Rightarrow det\ A=1`

`A^(-1)=(1)/(det\ A)\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}\\\Rightarrow A^(-1)=(1)/(1)\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}\\\Rightarrow A^(-1)=\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}`

`\therefore A^(-1)=\begin{bmatrix}1 & 2\\ -4 & -7\end{bmatrix}`

`A.A^(-1)=\begin{bmatrix}-7 & -2\\ 4 & 1\end{bmatrix}* \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}`

Hence, confirmed