Question

With $1,\omega ,{\omega }^2$ as cube roots of unity, inverse of which of the following matrices exists

• A. $\left[\begin{array}{cc}1& \omega \\ \omega & {\omega }^2\end{array}\right]$
• B. $\left[\begin{array}{cc}{\omega }^2& 1\\ 1& \omega \end{array}\right]$
• C. $\left[\begin{array}{cc}\omega & {\omega }^2\\ {\omega }^2& 1\end{array}\right]$
• D. None of these

Answer 1

Answer: (D)

None of these

The inverse of a matrix exists if its determinant is not equal $0$.
For option A,
Let $A=\left[\begin{array}{cc}1& \omega \\ \omega & {\omega }^2\end{array}\right]$
Here, $|A|=0$
Hence, inverse does not exists.
For option B,
Let $A=\left[\begin{array}{cc}{\omega }^2& 1\\ 1& \omega \end{array}\right]$
Here, $|A|=0$
Hence, inverse does not exists.
For option C,
Let $A=\left[\begin{array}{cc}\omega & {\omega }^2\\ {\omega }^2& 1\end{array}\right]$
Here, $|A|=0$
Hence, inverse does not exists.
Hence, the inverse does not exist for any of the given matrices