Consider the matrices \(A = \begin{bmatrix}9 & 0 & 0\\0 & 16 & 0\\0 & 0 & 25\end{bmatrix}\) and \(B = \begin{bmatrix}1/5 & 0 & 0\\0 & 1/4 & 0\\0 & 0 & 1/3\end{bmatrix}\). The value of \(|(AB)^{-1}|\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{60}\)
The determinant of a diagonal matrix is the product of its diagonal entries:
\(|A| = 9 \times 16 \times 25 = 3600\)
\( \displaystyle |B| = \frac{1}{5} \times \frac{1}{4} \times \frac{1}{3} = \frac{1}{60} \)
Using \(|AB| = |A||B|\) and \(|M^{-1}| = \frac{1}{|M|}\):
\( \displaystyle |AB| = 3600 \times \frac{1}{60} = 60 \)
\( \displaystyle |(AB)^{-1}| = \frac{1}{|AB|} = \frac{1}{60} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{15}{212}\) — Incorrect. The value is \(\frac{1}{60}\), not \(\frac{15}{212}\).
Option (2) 60 — Incorrect. 60 is \(|AB|\) itself; the inverse has determinant \(\frac{1}{|AB|}\).
Option (3) \(\displaystyle\frac{212}{15}\) — Incorrect. The value is \(\frac{1}{60}\), not \(\frac{212}{15}\).