Question

Find the inverse of the following matrix:

\[\begin{bmatrix}2 & 5 \\ - 3 & 1\end{bmatrix}\]

Answer 1

\[ D = \begin{bmatrix}2 & 5 \\ - 3 & 1\end{bmatrix}\]
\[\left| D \right| = 2 + 15 = 17 \neq 0\]
D is a singular matrix; therefore, it is invertible .
\[\text{ Let }C_{ij}\text{ be a cofactor of  }d_{ij}\text{ in D. }\]
Now, 
\[ C_{11} = 1 \]
\[ C_{12} = 3\]
\[ C_{21} = - 5\]
\[ C_{22} = 2\]
\[adjD = \begin{bmatrix}1 & 3 \\ - 5 & 2\end{bmatrix}^T = \begin{bmatrix}1 & - 5 \\ 3 & 2\end{bmatrix}\]
\[ \therefore D^{- 1} = \frac{1}{\left| D \right|}adjD = \frac{1}{17}\begin{bmatrix}1 & - 5 \\ 3 & 2\end{bmatrix}\]