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Question
If \[A = \begin{pmatrix}\cos\alpha & - \sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix},\] ,find adj·A and verify that A(adj·A) = (adj·A)A = |A| I3.
Solution
Consider the matrix,
\[A = \begin{pmatrix}\cos\alpha & - \sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix},\] \[\text { adj }\left( A \right) = C^T\] Where, C is cofactor matrix.
\[C = \begin{pmatrix}\cos \alpha & - \sin\alpha & 0 \\ \sin \alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix}, \]
\[\text { Adj }\left( A \right) = C^T = \begin{pmatrix}cos\alpha & sin\alpha & 0 \\ - sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix}\]
Now,
\[A . \text { Adj }\left( A \right) = \begin{pmatrix}cos\alpha & - sin\alpha & 0 \\ sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}cos\alpha & sin\alpha & 0 \\ - sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix}\]
\[ = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} . . . . . (1)\]
\[\text { Adj }\left( A \right) . A = \begin{pmatrix}cos\alpha & sin\alpha & 0 \\ - sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}cos\alpha & - sin\alpha & 0 \\ sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix}\]
\[ = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} . . . . . (2)\]
\[\left| A \right| = \begin{vmatrix}cos\alpha & - sin\alpha & 0 \\ sin\alpha & cos\alpha & 0 \\ 0 & 0 & 1\end{vmatrix}\]
\[ = cos\alpha\left( cos\alpha - 0 \right) + sin\alpha\left( sin\alpha - 0 \right) + 0\]
\[ = 1 . . . . . (3)\]
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