. $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&3\\{ - 4}&{ - 6}\end{array}} \right]$

Let $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&3\\{ - 4}&{ - 6}\end{array}} \right]$

$|A| = - 12 + 12 \Rightarrow |A| = 0$

Let ${A_{ij}}$ be co-factors of ${a_{ij}}$ in A.

Then, the co-factors of elements of A are given by

${A_{11}} = {( - 1)^{1 + 1}}( - 6) = - 6,{A_{12}} = {( - 1)^{1 + 2}}( - 4) = 4$

${A_{21}} = {( - 1)^{2 + 1}}(3) = - 3,{A_{22}} = {( - 1)^{2 + 2}}(2) = 2$

% $\Rightarrow$ $adjA = {\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 6}&4\\{ - 3}&2\end{array}} \right]^{\prime}} = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 6}&{ - 3}\\4&2\end{array}} \right]$

$\therefore$ $A(adjA) = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&3\\{ - 4}&{ - 6}\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 6}&{ - 3}\\4&2\end{array}} \right] = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&0\\0&0\end{array}} \right]$

$(adjA)/A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 6}&{ - 3}\\4&2\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&3\\{ - 4}&{ - 6}\end{array}} \right] = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&0\\0&0\end{array}} \right]$ and $|A|I = O$ as $|A| = 0$

Hence, $A(adjA) = (adjA)A = |A|I$