Find the cofactors of the elements in the determinant $\left|\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\end{array}\right|$ by Direct Method.

Let $∆=\left|\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\end{array}\right|$

$Step I$ Write down the three rows of the $∆$ and rewrite first two rows.

i.e.,             $\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\\ 1& 2& 3\\ -4& 3& 6\end{array}$

$step II$  After step I, rewrite first two columns

i.e.,      $\begin{array}{ccc}1& 2& 3\\ -4& 3& 6\\ 2& -7& 9\\ 1& 2& 3\\ -4& 3& 6\end{array}\begin{array}{cc}1& 2\\ -4& 3\\ 2& -7\\ 1& 2\\ -4& 3\end{array}$

$Step III$  After Step II, deleting first row and first column, then we get all cofactors i.e.,

 or      ${∆}^c=\left|\begin{array}{ccc}69& 48& 22\\ -39& 3& 11\\ 3& -18& 11\end{array}\right|$