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. $\left[\begin{array}{ccc}9& 1& 4\\ -3& 4& 5\\ 5& -3& -1\end{array}\right]$
• B. $\left[\begin{array}{ccc}9& -1& -4\\ 3& 4& 5\\ 5& 3& -1\end{array}\right]$
• C. $\left[\begin{array}{ccc}-9& -1& -4\\ -3& -4& 5\\ 5& -3& -1\end{array}\right]$
• 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 co-factor of $a_{ij}$ in A. Then the co-factor of elements of A are given by

$c_{11}=\left|\begin{array}{cc}1& -3\\ 2& 3\end{array}\right|=9$

$c_{12}=(-1)\left|\begin{array}{cc}2& -3\\ -1& 3\end{array}\right|=-3$

$c_{13}=\left|\begin{array}{cc}2& 1\\ -1& 2\end{array}\right|=5$

$c_{21}=(-1)\left|\begin{array}{cc}1& 1\\ 2& 3\end{array}\right|=-1$

$c_{22}=\left|\begin{array}{cc}1& 1\\ -1& 3\end{array}\right|=4$

$c_{23}=(-1)\left|\begin{array}{cc}1& 1\\ -1& 2\end{array}\right|=-3$

$c_{31}=\left|\begin{array}{cc}1& 1\\ 1& -3\end{array}\right|=-4$

$c_{32}=(-1)\left|\begin{array}{cc}1& 1\\ 2& -3\end{array}\right|=5$

$c_{33}=\left|\begin{array}{cc}1& 1\\ 2& 1\end{array}\right|=-1$

$\therefore adjA={\left[\begin{array}{ccc}9& -3& 5\\ -1& 4& -3\\ -4& 5& -1\end{array}\right]}^T=\left[\begin{array}{ccc}9& -1& -4\\ -3& 4& 5\\ 5& -3& -1\end{array}\right]$