$\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2\\3&4\end{array}} \right]$

Let $P = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2\\3&4\end{array}} \right]$ . Let ${A_{ij}}$ be cofactors of ${a_{ij}}$ in P.

Then, the cofactors of elements of P are given by

${A_{11}} = {( - 1)^{1 + 1}}(4) = 4$,
${A_{12}} = {( - 1)^{1 + 2}}(3) = - 3,$

${A_{21}} = {( - 1)^{2 + 1}}(2) = - 2$

${A_{22}} = {( - 1)^{2 + 2}}(1) = 1$

% $\therefore$ $adj$ $P = {\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}4&{ - 3}\\{ - 2}&1\end{array}} \right]^{\prime}} = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}4&{ - 2}\\{ - 3}&1\end{array}} \right]$