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Definition of a Determinant

Evaluate co...

Question

Evaluate $∣ ∣ ∣ ∣ \begin{matrix} cos α cos β & cos α sin β & - sin α - sin β & cos β & 0 sin α cos β & sin α sin β & cos α \end{matrix} ∣ ∣ ∣ ∣$

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Solution

$∣ ∣ ∣ ∣ \begin{matrix} cos α cos β & cos α sin β & - sin α - sin β & cos β & 0 sin α cos β & sin α sin β & cos α \end{matrix} ∣ ∣ ∣ ∣$

$= cos α cos β (cos α cos β) + cos α sin β (cos α sin β) + sin α (sin α {sin}^{2} β + sin α {cos}^{2} β)$

$= {cos}^{2} α {cos}^{2} β + {cos}^{2} α {sin}^{2} β + {sin}^{2} α (1)$

$= {cos}^{2} (α) [{cos}^{2} β + {sin}^{2} β] + {sin}^{2} α$

$= {cos}^{2} α + {sin}^{2} α$
$= 1$

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