Question
Find adjoint of the matrix $\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|$
$\Rightarrow \left| A \right| = \left| {\begin{array}{*{20}{c}} 1&2 \\ 3&4 \end{array}} \right|$
$\therefore$ A11 = Cofactor of ${a_{11}} = {\left( { - 1} \right)^2}\left( 4 \right) = 4$
A12 = Cofactor of ${a_{12}} = {\left( { - 1} \right)^3}\left( 3 \right) = - 3$
A21 = Cofactor of ${a_{21}} = {\left( { - 1} \right)^3}\left( 2 \right) = - 2$
A22 = Cofactor of ${a_{22}} = {\left( { - 1} \right)^4}\left( 1 \right) = 1$
$\therefore adjA = \left| {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}} \\ {{A_{21}}}&{{A_{22}}} \end{array}} \right|$
$= \left| {\begin{array}{*{20}{c}} 4&{ - 3} \\ { - 2}&1 \end{array}} \right|$
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$\begin{bmatrix}-1&5&6\\ \sqrt{3}&5&6\\2&3&-1\end{bmatrix}$