Discussion
If A and B are square matrices of order 3 such that \(|A| = -1\), \(|B| = 3\), then \(|3AB|\) equals
Answer: A
The determinant is -81.
For square matrices of order n, \(|kA| = k^{n}|A|\) and \(|AB| = |A||B|\). Here n = 3:
\[ \begin{aligned} |3AB| &= 3^{3}\,|A|\,|B| \\ &= 27 \times (-1) \times 3 = -81 \end{aligned} \]
-9 comes from multiplying the determinant by 3 instead of \(3^{3}\), as if the scalar multiplied only one row. -27 leaves out |B|. 81 drops the minus sign of |A|.
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