Question

The inverse matrix of $\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$, is

• A. $\left[\begin{array}{ccc}\frac{1}{2}& -\frac{1}{2}& \frac{1}{2}\\ -4& 3& -1\\ \frac{5}{2}& -\frac{3}{2}& \frac{1}{2}\end{array}\right]$
• B. $\left[\begin{array}{ccc}\frac{1}{2}& -4& \frac{5}{2}\\ 1& -6& 3\\ 1& 2& -1\end{array}\right]$
• C. $\frac{1}{2}\left[\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\\ 4& 2& 3\end{array}\right]$
• D. $\frac{1}{2}\left[\begin{array}{ccc}1& -1& -1\\ -8& 6& -2\\ 5& -3& 1\end{array}\right]$

Answer 1

Answer: (A)

$\left[\begin{array}{ccc}\frac{1}{2}& -\frac{1}{2}& \frac{1}{2}\\ -4& 3& -1\\ \frac{5}{2}& -\frac{3}{2}& \frac{1}{2}\end{array}\right]$

Let $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$
$\therefore \left|A\right|=0-1(1-9)+2(1-6)$
$=8-10=-2\ne 0$
$adj\left(A\right)=\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$
$\therefore A^{-1}=\frac{adj\left(A\right)}{\left|A\right|}$
$=\frac{1}{-2}\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$
$=\left[\begin{array}{ccc}\frac{1}{2}& \frac{-1}{2}& \frac{1}{2}\\ -4& 3& -1\\ \frac{5}{2}& \frac{-3}{2}& \frac{1}{2}\end{array}\right]$