. $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 1}&5\\{ - 3}&2\end{array}} \right]$

Let $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 1}&5\\{ - 3}&2\end{array}} \right]$

Then, $|A| = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 1}&5\\{ - 3}&2\end{array}} \right| = - 2 + 15 = 13 \ne 0.$

So, A is a non-singular matrix and therefore, it is invertible.

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

the cofactors of elements of A are given by

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

${A_{21}} = {( - 1)^{2 + 1}}(5) = - 5,{A_{22}} = {( - 1)^{2 + 2}}( - 1) = - 1$

% $\therefore$ $adjA = {\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&3\\{ - 5}&{ - 1}\end{array}} \right]^{\prime}}$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 5}\\3&{ - 1}\end{array}} \right]$

Hence, ${A^{ - 1}} = \cfrac{1}{{|A|}}(adj\,\;A) = \cfrac{1}{{13}}\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 5}\\3&{ - 1}\end{array}} \right]$