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  4. If A=[ cos θ - sin θ sin θ cos θ] then ( text adj A)-1 is equal to

Q. If $A=\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}$ then $(\text { adj } A)^{-1}$ is equal to

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Solution:

$A =\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}$ adj $A=\begin{bmatrix}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{bmatrix}$
$|\text{adj} A|=\cos ^{2} \theta+\sin ^{2} \theta=1$
$\text{adj}(\text{adj} A)=\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}$
$\therefore(\text{adj} A)^{-1}=\frac{\text{adj}(\text{adj} A)}{|\text{adj}(A)|}=\frac{1}{1}\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}=A$

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