Question

The inverse of the matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ a& 1& 0\\ b& c& 1\end{array}\right]$ is

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

Answer 1

The correct option is A $\left[\begin{array}{ccc}1& 0& 0\\ -a& 1& 0\\ ac-b& -c& 1\end{array}\right]$

$A=\left[\begin{array}{ccc}1& 0& 0\\ a& 1& 0\\ b& c& 1\end{array}\right]$
$\text{Co-factor matrix of}A=C=\left[\begin{array}{ccc}1& -a& ac-b\\ 0& 1& -c\\ 0& 0& 1\end{array}\right]$
$\text{adj}A=C^T=\left[\begin{array}{ccc}1& 0& 0\\ -a& 1& 0\\ ac-b& -c& 1\end{array}\right]$

$\Rightarrow \left|A\right|=1\Rightarrow A^{-1}=\frac{adjA}{|A|}=\left[\begin{array}{ccc}1& 0& 0\\ -a& 1& 0\\ ac-b& -c& 1\end{array}\right]$