Question

If $A=\left[\begin{array}{ccc}P& 0& 0\\ 0& Q& 0\\ 0& 0& R\end{array}\right],$ where $P,Q$ and $R$ are non-zero real numbers, then $A^{-1}$ equals

• A. $\frac{1}{PQR}{I}_3$
• B. $\frac{1}{PQR}A$
• C. $\frac{1}{PQR}\left[\begin{array}{ccc}1/P& 0& 0\\ 0& 1/Q& 0\\ 0& 0& 1/R\end{array}\right]$
• D. $\left[\begin{array}{ccc}1/P& 0& 0\\ 0& 1/Q& 0\\ 0& 0& 1/R\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{ccc}1/P& 0& 0\\ 0& 1/Q& 0\\ 0& 0& 1/R\end{array}\right]$

Given matrix, $A=\left|\begin{array}{ccc}P& 0& 0\\ 0& Q& 0\\ 0& 0& R\end{array}\right|$

$\left|A\right|=PQR$ and

$adjA=\left|\begin{array}{ccc}QR& 0& 0\\ 0& PR& 0\\ 0& 0& PQ\end{array}\right|$

$\therefore A^{-1}=\frac{1}{\left|A\right|}\left(adjA\right)$

$\Rightarrow A^{-1}=\frac{1}{PQR}\left|\begin{array}{ccc}QR& 0& 0\\ 0& PR& 0\\ 0& 0& PQ\end{array}\right|=\left|\begin{array}{ccc}\frac{1}{P}& 0& 0\\ 0& \frac{1}{Q}& 0\\ 0& 0& \frac{1}{R}\end{array}\right|$