If k>I, and the determinant of the matrix A² , where A = $[\begin{matrix} k & k a & a \\ 0 & a & k a \\ 0 & 0 & k \end{matrix}]$ , is k²then $|\begin{matrix} a \end{matrix}| =$

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$k^{2}$

$\frac{1}{k}$

$\frac{1}{k^{2}}$

answer is D.

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Detailed Solution

$|A| = |\begin{matrix} k & k a & a \\ 0 & a & k a \\ 0 & 0 & k \end{matrix}| = k^{2} a {|A|}^{2} = k^{4} a^{2} = k^{2} \Rightarrow a^{2} = \frac{1}{k^{2}} ∴ |a| = \frac{1}{k}$

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