Question

If $\left|\begin{array}{ccc}a& b& 0\\ 0& a& b\\ b& 0& a\end{array}\right|=0$, then

• A. $a$ is one of teh cube roots of unity
• B. $b$ is one of the cube roots of unity
• C. $\left(\frac{a}{b}\right)$ is one of the cube roots of unity
• D. $\left(\frac{a}{b}\right)$ is one of the cube roots of $-1$

Answer 1

The correct option is D $\left(\frac{a}{b}\right)$ is one of the cube roots of $-1$

Given : $\left|\begin{array}{ccc}a& b& 0\\ 0& b& a\end{array}\right|=0$

On solving the determinant, we get

$\to a^3+b^3=0$

$\to \frac{a^3}{b^3}+1=0$

$\to {\left(\frac{a}{b}\right)}^3=-1$

$\left(\frac{a}{b}\right)$ is the cube root of using.