Question

The adjoint of the matrix $A=\left[\begin{array}{ccc}1& 1& 1\\ 2& 1& -3\\ -1& 2& 3\end{array}\right]$ is

• A. $\frac{1}{11}\left[\begin{array}{ccc}9& -1& -4\\ -3& 4& 5\\ 5& -3& -1\end{array}\right]$
• B. $\left[\begin{array}{ccc}9& 1& -4\\ 3& 4& -5\\ 5& 3& -1\end{array}\right]$
• C. $\left[\begin{array}{ccc}9& -3& 5\\ -1& 4& -3\\ -4& 5& -1\end{array}\right]$
• D. $\left[\begin{array}{ccc}9& -1& -4\\ -3& 4& 5\\ 5& -3& -1\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{ccc}9& -1& -4\\ -3& 4& 5\\ 5& -3& -1\end{array}\right]$

Let $C_{ij}$ be a cofactor of $a_{ij}$ in A. Then, the cofactors of elements of A are given by
$C_{11}=\left[\begin{array}{cc}1& -3\\ 2& 3\end{array}\right]=9$
$C_{12}=-\left[\begin{array}{cc}2& -3\\ -1& 3\end{array}\right]=-3$
$C_{13}=\left[\begin{array}{cc}2& 1\\ -1& 2\end{array}\right]=5$
$C_{21}=-\left[\begin{array}{cc}1& 1\\ 2& 3\end{array}\right]=-1$
$C_{22}=\left[\begin{array}{cc}1& 1\\ -1& 3\end{array}\right]=4$
$C_{23}=-\left[\begin{array}{cc}1& 1\\ -1& 2\end{array}\right]=-3$
$C_{31}=\left[\begin{array}{cc}1& 1\\ 1& -3\end{array}\right]=-4$
$C_{32}=-\left[\begin{array}{cc}1& 1\\ 2& -3\end{array}\right]=5$
$C_{33}=\left[\begin{array}{cc}1& 1\\ 2& 1\end{array}\right]=-1$
$\therefore adjA={\left[\begin{array}{ccc}9& -3& 5\\ -1& 4& -3\\ -4& 5& -1\end{array}\right]}^{\prime }=\left[\begin{array}{ccc}9& -1& -4\\ -3& 4& 5\\ 5& -3& -1\end{array}\right]$