Question

If $A=\left[\begin{array}{ccc}0& \sin \alpha & \sin \alpha \sin \beta \\ -\sin \alpha & 0& \cos \alpha \sin \beta \\ -\sin \alpha \sin \beta & -\cos \alpha \cos \beta & 0\end{array}\right]$, then which of the following is true?

• A. $|A|$ is independent of $\alpha$ and $\beta$.
• B. $A^{-1}$ depends only on $\alpha$.
• C. $A^{-1}$ depends only on $\beta$.
• D. None of these

Answer 1

Answer: (A)

$|A|$ is independent of $\alpha$ and $\beta$.

$A=\left[\begin{array}{ccc}0& \sin \alpha & \sin \alpha \sin \beta \\ -\sin \alpha & 0& \cos \alpha \cos \beta \\ -\sin \alpha \sin \beta & -\cos \alpha \cos \beta & 0\end{array}\right]$

A is a skew symmetric matrix

$\Rightarrow \left|A\right|=0$

$\Rightarrow A^{-1}$ does not exist and

$\left|A\right|$ is independent of $\alpha \&\beta$

Option A