The value of the determinant $\left| \begin{array}{r} \cos{\alpha\ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \sin{\alpha\ }\ \ \ \ 1 \ \sin{\alpha\ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos{\alpha\ }\ \ \ \ \ 1 \ \cos(\alpha + \beta) - \sin(\alpha + \beta)\ 1 \end{array} ight|$is

Question

The value of the determinant $\left| \begin{array}{r} \cos{\alpha\ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \sin{\alpha\ }\ \ \ \ 1 \ \sin{\alpha\ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos{\alpha\ }\ \ \ \ \ 1 \ \cos(\alpha + \beta) - \sin(\alpha + \beta)\ 1 \end{array} ight|$is

Dr. Abubacker · Accepted Answer

(a)
Applying $R_{3} ightarrow R_{3} - R_{1}\left( \cos\beta ight) + R_{2}(\sin\beta)$
$\left| \begin{matrix}
\cos\alpha & - \sin\alpha & 1 \
\sin\alpha & \cos\alpha & 1 \
0 & 0 & 1 + \sin{\beta - \cos\beta}
\end{matrix} ight|$
$= \left( 1 + \sin\beta - \cos\beta ight)(\cos^{2}{\alpha + \sin^{2}{\alpha)}}$
$= 1 + \sin\beta - \cos{\beta,\ }$ which is independent of $\alpha$

Q. The value of the determinant $\left| \begin{array}{r} \cos{\alpha\ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \sin{\alpha\ }\ \ \ \ 1 \\ \sin{\alpha\ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos{\alpha\ }\ \ \ \ \ 1 \\ \cos(\alpha + \beta) - \sin(\alpha + \beta)\ 1 \end{array} \right|$is

Explanation

Reviewed by Dr. Abubacker

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