Question

$Adj\left[\begin{array}{ccc}1& 0& 2\\ -1& 1& -2\\ 0& 2& 1\end{array}\right]=\left[\begin{array}{ccc}5& a& -2\\ 1& 1& 0\\ -2& -2& b\end{array}\right]\Rightarrow \left[ab\right]=$ ____

• A. $\left[\begin{array}{cc}-4& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}-4& -1\end{array}\right]$
• C. $\left[\begin{array}{cc}4& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}4& -1\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{cc}4& 1\end{array}\right]$

Given, Adj$\left[\begin{array}{ccc}1& 0& 2\\ -1& 1& -2\\ 0& 2& 1\end{array}\right]=\left[\begin{array}{ccc}5& 9& -2\\ 1& 1& 0\\ -2& -2& 6\end{array}\right]$

Adjoint of matrix = Transpose of cofactor matrix

$C_{11}=\left|\begin{array}{cc}1& -2\\ 2& 1\end{array}\right|=5$

$C_{12}=\left|\begin{array}{cc}-1& -2\\ 0& 1\end{array}\right|=1$

$C_{13}=\left|\begin{array}{cc}-1& 1\\ 0& 2\end{array}\right|=-2$

$C_{21}=-\left|\begin{array}{cc}0& 2\\ 2& 1\end{array}\right|=4$

$C_{22}=\left|\begin{array}{cc}1& 2\\ 0& 1\end{array}\right|=1$

$C_{23}=-\left|\begin{array}{cc}1& 0\\ 0& 2\end{array}\right|=-2$

$C_{31}=\left|\begin{array}{cc}0& 2\\ 2& 1\end{array}\right|=-4$

$C_{32}=-\left|\begin{array}{cc}1& 2\\ -1& -2\end{array}\right|=0$

$C_{33}=\left|\begin{array}{cc}1& 0\\ -1& 1\end{array}\right|=1$

Cofactor matrix $C=\left[\begin{array}{ccc}5& 1& -2\\ 4& 1& -2\\ -4& 0& 1\end{array}\right]$

Adj matrix$=\left[\begin{array}{ccc}5& 4& -4\\ 4& 1& 0\\ -2& -2& 1\end{array}\right]=\left[\begin{array}{ccc}5& a& -2\\ 1& 1& 0\\ -2& -2& b\end{array}\right]$

$\left[\begin{array}{cc}a& b\end{array}\right]=\left[\begin{array}{cc}4& 1\end{array}\right]$