Question

Find the inverse of each of the matrices,if it exists.\(\begin{bmatrix}4&5\\3&4\end{bmatrix}\)

Answer 1

The correct answer is \(A^{-1}=\begin{bmatrix}4&-5\\-3&4\end{bmatrix}\)
Let \(A=\begin{bmatrix}4&5\\3&4\end{bmatrix}\) We know that \(A = IA\)
so \(\begin{bmatrix}4&5\\3&4\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}\)
\(\implies \begin{bmatrix}1&1\\3&4\end{bmatrix}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}A (R_1\rightarrow R_1-R_2)\)
\(\implies\begin{bmatrix}1&1\\0&1\end{bmatrix}=\begin{bmatrix}1&-1\\-3&4\end{bmatrix} A (R_2\rightarrow R_2-3R_1)\)
\(\implies\begin{bmatrix}1&1\\0&1\end{bmatrix}=\begin{bmatrix}4&-5\\3&4\end{bmatrix} A (R_1\rightarrow R_1-R_2)\)
therefore \(A^{-1}=\begin{bmatrix}4&-5\\-3&4\end{bmatrix}\)