Question-2

Suppose there exist three square matrices \(A,D,P\) of order \(3\) such that \(D=PAP^{-1}\) and \(D\) is diagonal. Find a relationship between \(|D|\) and \(|A|\). If \(D=I\), is it necessary that \(A\) is also equal to \(I\)?

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We have:

\[ \begin{aligned} |D| & =\left|PAP^{-1}\right|\\ & =|P|\cdot|A|\cdot\left|P^{-1}\right|\\ & =|P|\cdot|A|\cdot\cfrac{1}{|P|}\\ & =|A| \end{aligned} \]

Thus, the determinants of \(A\) and \(D\) are equal. We now turn to the second part of the question. If \(D=I\), then:

\[ \begin{aligned} I & =PAP^{-1}\\ P^{-1}IP & =A\\ P^{-1}P & =A\\ I & =A \end{aligned} \]

Thus, if \(D=I\), then \(A=I\).