Question

$A={\left[\begin{array}{ccc}P& 0& 0\\ 0& Q& 0\\ 0& 0& R\end{array}\right]}_{3\times 3}$ 'where P, Q, R are non zero real numbers then equals $A^{-1}$ 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 $\left[\begin{array}{ccc}1/P& 0& 0\\ 0& 1/Q& 0\\ 0& 0& 1/R\end{array}\right]$

For given matrix $A=\left[\begin{array}{ccc}P& 0& 0\\ 0& Q& 0\\ 0& 0& R\end{array}\right]$
Here, $\left|A\right|=PQR$
$\left(adjA\right)=\left[\begin{array}{ccc}QR& 0& 0\\ 0& PR& 0\\ 0& 0& PQ\end{array}\right]$
$\therefore A^{-1}=\frac{1}{\left|A\right|}\left(adjA\right)$
$=\frac{1}{PQR}\left[\begin{array}{ccc}QR& 0& 0\\ 0& PR& 0\\ 0& 0& PQ\end{array}\right]$
$=\left[\begin{array}{ccc}1/P& 0& 0\\ 0& 1/Q& 0\\ 0& 0& 1/R\end{array}\right]$