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  4. The adjoint of the matrix [ beginmatrix cos θ sin θ - sin θ cos θ endmatrix ] is

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Q. The adjoint of the matrix $ \left[ \begin{matrix} \cos \,\theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{matrix} \right] $ is

J & K CETJ & K CET 2008Determinants

Solution:

Let $ A=\left[ \begin{matrix} \cos \,\theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{matrix} \right] $
Here, $ {{C}_{11}}=\cos \theta ,\,{{C}_{12}}=\sin \theta $
$ {{C}_{21}}=-\sin \theta ,{{C}_{22}}=\cos \theta $
$ \therefore $ adj $ (A)=\left[ \begin{matrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{matrix} \right] $
$ =\left[ \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{matrix} \right] $