Question-3

Show that \(\text{det}(AB)\) is zero if and only if at least one of \(\text{det}(A)\) and \(\text{det}(B)\) is zero.

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If \(\text{det}(AB)=0\), then \(\text{det}(A)\text{det}(B)=0\). If the product of two numbers is zero, at least one of them has to be zero. Going the other way, if \(\text{det}(A)\) or \(\text{det}(B)\) is zero, \(\text{det}(AB)\) is zero.