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Question
Find the inverse of the following matrix:
Solution
\[B = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\]
\[\left| B \right| = 0 - 1 = - 1 \neq 0\]
B is a singular matrix; therefore, it is invertible .
\[\text{ Let }C_{ij}\text{ be a cofactor of }b_{ij}\text{ in B. }\]
Now,
\[ C_{11} = 0 \]
\[ C_{12} = - 1\]
\[ C_{21} = - 1\]
\[ C_{22} = 0\]
\[adjB = \begin{bmatrix}0 & - 1 \\ - 1 & 0\end{bmatrix}^T = \begin{bmatrix}0 & - 1 \\ - 1 & 0\end{bmatrix}\]
\[ \therefore B^{- 1} = \frac{1}{\left| B \right|}adjB = \frac{1}{- 1}\begin{bmatrix}0 & - 1 \\ - 1 & 0\end{bmatrix} = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\]
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