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

$\frac{1}{60}$

The correct answer is Option (4) → $\frac{1}{60}$

Given

$A=\begin{pmatrix}9&0&0\\0&16&0\\0&0&25\end{pmatrix},\quad B=\begin{pmatrix}\frac{1}{5}&0&0\\0&\frac{1}{4}&0\\0&0&\frac{1}{3}\end{pmatrix}$

$|A|=9\cdot16\cdot25=3600$

$|B|=\frac{1}{5}\cdot\frac{1}{4}\cdot\frac{1}{3}=\frac{1}{60}$

$|AB|=|A||B|=3600\times\frac{1}{60}=60$

$|(AB)^{-1}|=\frac{1}{|AB|}=\frac{1}{60}$

final answer: $\frac{1}{60}$