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If the matrix $A =\begin{bmatrix}\cos θ&\sin θ\\-\sin θ&\cos θ\end{bmatrix}$, then $A^2$ is equal to: |
$\begin{bmatrix}\cos 2θ&\sin 2θ\\-\sin 2θ&\cos 2θ\end{bmatrix}$ $\begin{bmatrix}\cos^2 θ&\sin^2 θ\\-\sin^2 θ&\cos^2 θ\end{bmatrix}$ $\begin{bmatrix}\cos θ^2&\sin θ^2\\-\sin θ^2&\cos θ^2\end{bmatrix}$ $\begin{bmatrix}\cos θ+\sin θ&\cos θ-\sin θ\\\sin θ-\cos θ&\cos θ+\sin θ\end{bmatrix}$ |
$\begin{bmatrix}\cos 2θ&\sin 2θ\\-\sin 2θ&\cos 2θ\end{bmatrix}$ |
$A =\begin{bmatrix}\cos θ&\sin θ\\-\sin θ&\cos θ\end{bmatrix}$ so, $A^2=A×A$ $=\begin{bmatrix}\cos θ&\sin θ\\-\sin θ&\cos θ\end{bmatrix}\begin{bmatrix}\cos θ&\sin θ\\-\sin θ&\cos θ\end{bmatrix}=\begin{bmatrix}\cos^2 θ-\sin^2 θ&2\cos θ\sin θ\\-2\cos θ\sin θ&\cos^2 θ-\sin^2 θ\end{bmatrix}$ $=\begin{bmatrix}\cos 2θ&\sin 2θ\\-\sin 2θ&\cos 2θ\end{bmatrix}$ $[∵\cos^2 θ-\sin^2 θ=\cos 2θ, 2\cos θ\sin θ=\sin 2θ]$ |
