Question-10

Suppose \(A\) and \(B\) are two \(3\times3\) matrices such that \(\text{det}(A)=4\) and \(B=3A\), find the value of \(\sqrt[3]{\text{det}(A^{2}B)}\).

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First:

\[ \begin{aligned} \text{det}(B) & =\text{det}(3A)\\ & =3^{3}\text{det}(A)\\ & =27\times4\\ & =108 \end{aligned} \]

Next:

\[ \begin{aligned} \text{det}(A^{2}B) & =\text{det}(A)^{2}\text{det}(B)\\ & =16\times108\\ & 1728 \end{aligned} \]

Finally, \(\sqrt[3]{1728}=12\).