Question

The inverse of $\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$ is:

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

Answer 1

Answer: (A)

$\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$

Let $A=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$
$\left|A\right|=-1\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=-1$
$Adj.A=\left[\begin{array}{ccc}0& -1& 0\\ -1& 0& 0\\ 0& 0& -1\end{array}\right]$
We have,
$A^{-1}=\frac{1}{\left|A\right|}.AdjA$
$\Rightarrow A^{-1}=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]=A$