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If $\left|\begin{array}{lll}a & b & 0 \\ 0 & a & b \\ b & 0 & a\end{array}\right|=0$, then which one of the following is correct?
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The correct answer is:
$\frac{a}{b}$ is one of the cube roots of $-1$.
$\left|\begin{array}{lll}\mathrm{a} & \mathrm{b} & 0 \\ 0 & \mathrm{a} & \mathrm{b} \\ \mathrm{b} & 0 & \mathrm{a}\end{array}\right|$
$=a\left[a^{2}-0\right]-b\left[-b^{2}\right]+0$
$=a^{3}+b^{3}=0$
$\Rightarrow a^{3}=-b^{3}$
$\Rightarrow\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{3}=-1$
Hence, $\frac{\mathrm{a}}{\mathrm{b}}$ is one of the cube roots of $-1$
$=a\left[a^{2}-0\right]-b\left[-b^{2}\right]+0$
$=a^{3}+b^{3}=0$
$\Rightarrow a^{3}=-b^{3}$
$\Rightarrow\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{3}=-1$
Hence, $\frac{\mathrm{a}}{\mathrm{b}}$ is one of the cube roots of $-1$
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