Question

Arrange the following matrices in ascending order of their determinant values.

$A$) $\left[\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right]$

$B)\left[\begin{array}{ccc}1& w& w^2\\ w& w^2& 1\\ w^2& 1& w\end{array}\right]$

$C\left)A\right(adjA)=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$, then $|A|$

$D)\left[\begin{array}{cc}0& i\\ -i& 0\end{array}\right]$

• A. $C,A,D,B$
• B. $D,B,A,C$
• C. $D,C,B,A$
• D. $C,D,B,A$

Answer 1

Answer: (B)

$D,B,A,C$

$A)$ $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$

$\Rightarrow |A|={\cos }^2\theta +{\sin }^2\theta =1$

$B)$ $B=\left[\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right]$

$\Rightarrow |B|=1({\omega }^3-1)-\omega ({\omega }^2-{\omega }^2)+{\omega }^2(\omega -{\omega }^4)=0$

$C)$ $A\left(adjA\right)=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

We know that $A\left(adjA\right)=|A|I$

$\Rightarrow \left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}|A|& 0\\ 0& |A|\end{array}\right]$

$\Rightarrow |A|=2$

$D)$ Let $A=\left[\begin{array}{cc}0& i\\ -i& 0\end{array}\right]$

$\Rightarrow |A|=0+i^2=-1$

So, the correct order is $D,B,A,C$.