Question

If $\omega$ is a complex cube root of unity and $A=\begin{pmatrix}\omega & 0\\0 & \omega\end{pmatrix}$, then $A^{-1}=$

• A. $A^2$
• B. $2A$
• C. $-A$
• D. $A$

Hint:

For cube roots of unity, remember $\omega^{-1}=\omega^2$.

Answer 1

The Correct Option is A

Step 1: Using property of cube roots of unity.
For complex cube roots of unity, \[ \omega^3=1 \Rightarrow \omega^{-1}=\omega^2 \]
Step 2: Finding the inverse of matrix $A$.
\[ A^{-1}=\begin{pmatrix}\omega^{-1} & 0\\0 & \omega^{-1}\end{pmatrix} =\begin{pmatrix}\omega^2 & 0\\0 & \omega^2\end{pmatrix} \]
Step 3: Expressing in terms of $A$.
\[ A^2=\begin{pmatrix}\omega^2 & 0\\0 & \omega^2\end{pmatrix} \]
Step 4: Conclusion.
\[ A^{-1}=A^2 \]