Question

The inverse of the matrix
$S=\left[\begin{array}{ccc}1& -1& 0\\ 1& 1& 1\\ 0& 0& 1\end{array}\right]\text{is}$

• A. $\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 0\\ 0& 1& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}0& 1& 1\\ -1& -1& 1\\ 1& 0& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}2& 2& -2\\ -2& 2& -2\\ 0& 2& 2\end{array}\right]$
• D. $\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& \frac{-1}{2}\\ \frac{-1}{2}& \frac{1}{2}& \frac{-1}{2}\\ 0& 0& 1\end{array}\right]$
  

Answer 1

The correct option is D $\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& \frac{-1}{2}\\ \frac{-1}{2}& \frac{1}{2}& \frac{-1}{2}\\ 0& 0& 1\end{array}\right]$


$S=\left[\begin{array}{ccc}1& -1& 0\\ 1& 1& 1\\ 0& 0& 1\end{array}\right]$
$\text{Determinant of matrix S}=\left|S\right|=1(1-0)+1(1-0)=2\ne 0\Rightarrow S^{-1}\text{exist}.$
Now, cof(S)=$\left[\begin{array}{ccc}1& -1& 0\\ 1& 1& 0\\ -1& -1& 2\end{array}\right]$

$\therefore adj\left(S\right)=\left[\begin{array}{ccc}1& 1& -1\\ -1& 1& -1\\ 0& 0& 2\end{array}\right]$
$\Rightarrow [S{]}^{-1}=\frac{adj\left(S\right)}{\left|S\right|}$=$\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& \frac{-1}{2}\\ \frac{-1}{2}& \frac{1}{2}& \frac{-1}{2}\\ 0& 0& 1\end{array}\right]$