Question

$S_1:1fA=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$ then $A^{-1}=\left[\begin{array}{ccc}1/a& 0& 0\\ 0& 1/b& 0\\ 0& 0& 1/c\end{array}\right]$
$S_2:$ lf $A==\left[\begin{array}{ccc}0& 0& a\\ 0& b& 0\\ c& 0& 0\end{array}\right]$ then $A^{-1}=\left[\begin{array}{ccc}0& 0& 1/a\\ 0& 1/b& 0\\ 1/c& 0& 0\end{array}\right]$

• A. only $S_1$ is true
• B. only $S_2$ is true
• C. both $S_1$ and $S_2$ are true
• D. Neither $S_1$ nor $S_2$

Answer 1

The correct option is A only $S_1$ is true

Statement 1:
Given , $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
Here, $|A|=abc$
$adjA=C^T={\left[\begin{array}{ccc}bc& 0& 0\\ 0& ac& 0\\ 0& 0& ab\end{array}\right]}^T$
$\Rightarrow adjA=\left[\begin{array}{ccc}bc& 0& 0\\ 0& ac& 0\\ 0& 0& ab\end{array}\right]$
$\Rightarrow A^{-1}=\left[\begin{array}{ccc}1/a& 0& 0\\ 0& 1/b& 0\\ 0& 0& 1/c\end{array}\right]$
Statement 2:
Given , $A=\left[\begin{array}{ccc}0& 0& a\\ 0& b& 0\\ c& 0& 0\end{array}\right]$
Here, $|A|=-abc$
$adjA=C^T={\left[\begin{array}{ccc}0& 0& -bc\\ 0& -ac& 0\\ 0& 0& -ab\end{array}\right]}^T$
$\Rightarrow adjA=\left[\begin{array}{ccc}0& 0& -bc\\ 0& -ac& 0\\ 0& 0& -ab\end{array}\right]$
$\Rightarrow A^{-1}=\left[\begin{array}{ccc}0& 0& 1/a\\ 0& 1/b& 0\\ 0& 0& 1/c\end{array}\right]$