Find the adjoint of the following matrices: C= & & Verify that (a… - Vidyadip

Question

Find the adjoint of the following matrices: $\text{C}=\begin{bmatrix} \cos\alpha & \sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}$Verify that (adjoint A) A = |A|I = A (adjoint A) for the above matrices.

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Answer

$\text{adjoint C}=\begin{bmatrix} \cos\alpha & -\sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}$
$\text{(adjoint C)C}=\begin{bmatrix}\cos^2\alpha-\sin^2\alpha & 0 \\ 0 & \cos^2\alpha-\sin^2\alpha \end{bmatrix}$
$|\text{C}|=\cos^2\alpha-\sin^2\alpha$
$|\text{C}|\text{I}=\begin{bmatrix}\cos^2\alpha-\sin^2\alpha & 0 \\ 0 & \cos^2\alpha-\sin^2\alpha \end{bmatrix}$
$\text{C(adjoint C)}=\begin{bmatrix}\cos^2\alpha-\sin^2\alpha & 0 \\ 0 & \cos^2\alpha-\sin^2\alpha \end{bmatrix}$
$\therefore\ \text{(adjoint C)}=|\text{C}|\text{I}=\text{C(adjoint C)}$
Hence verified.

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