Question

If $\left|\begin{array}{ccc}a& b& 0\\ 0& a& b\\ b& 0& a\end{array}\right|=0$, then which one of the following is correct?

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

Answer 1

Answer: (B)

$\frac{a}{b}$ is one of the cube roots of $-1$.

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

$\Rightarrow a\left(a^2\right)-b(-b^2)=0$

$\Rightarrow a^3=-b^3$

$=\frac{a}{b}=\sqrt[3]{3(-1)}$

Option B