Question

Find the inverse of ????5 3
2 4
????. Confirm your answer

Answer 1

`\therefore A^(-1)=\begin{bmatrix}0.29 & -0.21\\ -0.14 & 0.36\end{bmatrix}`

Explanation:

`A=\begin{bmatrix}5 & 3\\ 2 & 4\end{bmatrix}`

`det\ A=5* 4-3* 2\\\Rightarrow det\ A=14`

`A^(-1)=(1)/(det\ A)\begin{bmatrix}4 & -3\\ -2 & 5\end{bmatrix}\\\Rightarrow A^(-1)=(1)/(14)\begin{bmatrix}4 & -3\\ -2 & 5\end{bmatrix}\\\Rightarrow A^(-1)=\begin{bmatrix}(2)/(7) & (-3)/(14)\\ -(1)/(7) & (5)/(14)\end{bmatrix}`

`\therefore A^(-1)=\begin{bmatrix}0.29 & -0.21\\ -0.14 & 0.36\end{bmatrix}`

`A.A^(-1)=\begin{bmatrix}5\:&\:3\\ \:\:2\:&\:4\end{bmatrix}* \begin{bmatrix}(2)/(7)\:&\:(-3)/(14)\\ \:\:-(1)/(7)\:&\:(5)/(14)\end{bmatrix}\\\Rightarrow A.A^(-1)=\begin{pmatrix}5\cdot (2)/(7)+3\left(-(1)/(7)\right)&5\cdot (-3)/(14)+3\cdot (5)/(14)\\ 2\cdot (2)/(7)+4\left(-(1)/(7)\right)&2\cdot (-3)/(14)+4\cdot (5)/(14)\end{pmatrix}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}`

Hence, proved