$\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\3&0&{ - 2}\\1&0&3\end{array}} \right]$

Let $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\3&0&{ - 2}\\1&0&3\end{array}} \right]$

Let ${A_{ij}}$ be cofactor of ${a_{ij}}$ in A.

Then, the cofactors of elements of A are given by

${A_{11}} = {( - 1)^{1 + 1}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&{ - 2}\\0&3\end{array}} \right] = 0$

${A_{12}} = {( - 1)^{1 + 2}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&{ - 2}\\1&3\end{array}} \right] = - (9 + 2) = - 11$

${A_{13}} = {( - 1)^{1 + 3}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&0\\1&0\end{array}} \right] = 0$

${A_{21}} = {( - 1)^{2 + 1}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 1}&2\\0&3\end{array}} \right] = - ( - 3 - 0) = 3$

${A_{22}} = {( - 1)^{2 + 2}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2\\1&3\end{array}} \right] = 3 - 2 = 1$

${A_{23}} = {( - 1)^{2 + 3}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}\\1&0\end{array}} \right] = - (0 + 1) = - 1$

${A_{31}} = {( - 1)^{3 + 1}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 1}&2\\0&{ - 2}\end{array}} \right] = 2$

${A_{32}} = {( - 1)^{3 + 2}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2\\3&{ - 2}\end{array}} \right] = - ( - 2 - 6) = 8$

${A_{33}} = {( - 1)^{3 + 3}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}\\3&0\end{array}} \right] = 0 + 3 = 3$

$\therefore$ $adjA = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&{ - 11}&0\\3&1&{ - 1}\\2&8&3\end{array}} \right] = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&3&2\\{ - 11}&1&8\\0&{ - 1}&3\end{array}} \right]$

Now, $A(adjA) = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\3&0&{ - 2}\\1&0&3\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&3&2\\{ - 11}&1&8\\0&{ - 1}&8\end{array}} \right]$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{0 + 11 + 0}&{3 - 1 - 2}&{2 - 8 + 6}\\{0 + 0 + 0}&{9 + 0 + 2}&{6 + 0 - 6}\\{0 + 0 + 0}&{3 + 0 - 3}&{2 + 0 + 9}\end{array}} \right]$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{11}&0&0\\0&{11}&0\\0&0&{11}\end{array}} \right] = 11\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&0\\0&1&0\\0&0&1\end{array}} \right] = 11I$

$(AdjA)A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&3&2\\{ - 11}&1&8\\0&{ - 1}&3\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\3&0&{ - 2}\\1&0&3\end{array}} \right]$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{11}&0&0\\0&{11}&0\\0&0&{11}\end{array}} \right] = 11\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&0\\0&1&0\\0&0&1\end{array}} \right] = 11I$

Also, $|A| = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 1}&2\\3&0&{ - 2}\\1&0&3\end{array}} \right| = 1(0) + 1(9 + 2) + 2(0) = 11$

Hence, $A(AdjA) = (AAdjA)A = 11I = |A|I.$

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