Discussion
Let \(A\) be a 3×3 matrix and let B=3\(A\) . If |A|=5 , then the value of \(\dfrac{|\text{adj } B|}{|3A|}\) is equal to
Answer: D
The value is 135.
For a 3×3 matrix, \(|kA| = k^{3}|A|\) and \(|\text{adj } B| = |B|^{2}\).
\[ \begin{aligned} |B| &= |3A| = 27 \times 5 = 135 \\ \frac{|\text{adj } B|}{|3A|} &= \frac{135^{2}}{135} = 135 \end{aligned} \]
Using \(|3A| = 3|A| = 15\) is the usual slip (the factor 3 multiplies all three rows), and 125 or 25 come from working with \(|A|\) alone.
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