Question

If $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$, then $A^{-1}$ equals

• A. $\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
• B. $\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& ab& 0\\ 0& 0& ac\end{array}\right]$
• C. $\left[\begin{array}{ccc}1/a& 0& 0\\ 0& 1/b& 0\\ 0& 0& 1/c\end{array}\right]$
• D. $\left[\begin{array}{ccc}-a& 0& 0\\ 0& -b& 0\\ 0& 0& -c\end{array}\right]$

Answer 1

Answer: (C)

$\left[\begin{array}{ccc}1/a& 0& 0\\ 0& 1/b& 0\\ 0& 0& 1/c\end{array}\right]$

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