Given,
The matrix $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 5\\ 1& -1& -1\\ 2& 3& -1\end{array}\right]$

The determinant of A, |A| = 1(1 + 3) – 2( – 1 + 2) + 5(3 + 2) = 4 – 2 + 25 = 27 ≠ 0

This indicates that the matrix have inverse

In order to find adjoint of the matrix A let's find the minors of the matrix A $=\left[\begin{array}{ccc}\left|\begin{array}{cc}-1& -1\\ 3& -1\end{array}\right|& \left|\begin{array}{cc}1& -1\\ 2& -1\end{array}\right|& \left|\begin{array}{cc}1& -1\\ 2& 3\end{array}\right|\\ \left|\begin{array}{cc}2& 5\\ 3& -1\end{array}\right|& \left|\begin{array}{cc}1& 5\\ 2& -1\end{array}\right|& \left|\begin{array}{cc}1& 2\\ 2& 3\end{array}\right|\\ \left|\begin{array}{cc}2& 5\\ -1& -1\end{array}\right|& \left|\begin{array}{cc}1& 5\\ 1& -1\end{array}\right|& \left|\begin{array}{cc}1& 2\\ 1& -1\end{array}\right|\end{array}\right]$

Co-factors

$=\left[\begin{array}{ccc}(-1{)}^{1+1}(4)& (-1{)}^{1+2}(1)& (-1{)}^{1+3}(5)\\ (-1{)}^{2+1}(-17)& (-1{)}^{2+2}(-11)& (-1{)}^{2+3}(-1)\\ (-1{)}^{3+1}(3)& (-1{)}^{3+2}(-6)& (-1{)}^{3+3}(-3)\end{array}\right]=\left[\begin{array}{ccc}4& -1& 5\\ 17& -11& 1\\ 3& 6& -3\end{array}\right]$

Transpose of the matrix A with co-factors

${{\mathrm{C}}_{\mathrm{A}}}^{\mathrm{T}}={\left[\begin{array}{ccc}4& -1& 5\\ 17& -11& 1\\ 3& 6& -3\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{ccc}4& 17& 3\\ -1& -11& 6\\ 5& 1& -3\end{array}\right]=\mathrm{adj}\mathrm{A}$

The inverse of the matrix $\mathrm{A},{\mathrm{A}}^{-1}=\frac{1}{\left|\mathrm{A}\right|}\text{ adj }\mathrm{A}$

$=\frac{1}{27}\left[\begin{array}{ccc}4& 17& 3\\ -1& -11& 6\\ 5& 1& -3\end{array}\right]$