Question

prove that the determinant is independent of theta

Answer 1

$\textbf{Given:}$

$\left|\begin{array}{ccc}x&sin\theta&cos\theta\\-sin\theta&-x&1\\cos\theta&1&x\end{array}\right|$

$\textbf{To prove: The determinant is independent of \theta }$

$\text{Consider}$

$\left|\begin{array}{ccc}x&sin\theta&cos\theta\\-sin\theta&-x&1\\cos\theta&1&x\end{array}\right|$

$\text{Expanding along first row, we get}$

$=x(-x^2-1)-sin\theta(-x\;sin\theta-cos\theta)+cos\theta(-sin\theta+x\;cos\theta)$

$=-x^3-x+x\;sin^2\theta+sin\theta\;cos\theta-sin\theta\;cos\theta+x\;cos^2\theta$

$=-x^3-x+x\;sin^2\theta+x\;cos^2\theta$

$=-x^3-x+x(sin^2\theta+cos^2\theta)$

$=-x^3-x+x(1)$

$=-x^3-x+x$

$=-x^3\;\text{which is independent of \theta }$

$\textbf{Hence proved}$

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