Question

If $\text{Adj}\left[\begin{array}{ccc}1& 0& 2\\ -1& 1& -2\\ 0& 2& 1\end{array}\right]=\left[\begin{array}{ccc}5& a& -2\\ 1& 1& 0\\ -2& -2& b\end{array}\right],\text{then}\left[ab\right]=$

• A. $[-4-1]$
• B. $[-41]$
• C. $[4-1]$
• D. $\left[41\right]$

Answer 1

The correct option is D $\left[41\right]$

Let $A=\left[\begin{array}{ccc}1& 0& 2\\ -1& 1& -2\\ 0& 2& 1\end{array}\right]$
We know that $\text{Adj}A=C^T$
where $C=$Co-factor matrix of $A$
$C=\left[\begin{array}{ccc}1-(-4)& -(-1)& -2-0\\ -(0-4)& 1-0& -(2-0)\\ 0-2& -(-2+2)& 1-0\end{array}\right]$
$=\left[\begin{array}{ccc}5& 1& -2\\ 4& 1& -2\\ -2& 0& 1\end{array}\right]$
Now $C^T=\left[\begin{array}{ccc}5& 4& -2\\ 1& 1& 0\\ -2& -2& 1\end{array}\right]=\text{Adj}A$
$\Rightarrow a=4,b=1$