Examine the consistency of the system of equations.

The given system of equations is:

$3 x-y-2 z=2$

$2 y-z=-1$

$3 x-5 y=3$

This system of equations can be written in the form of AX = B, where

$A=\left[\begin{array}{ccc}3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right], X=\left[\begin{array}{c}x \\ y \\ z\end{array}\right]$ and $B=\left[\begin{array}{c}2 \\ -1 \\ 3\end{array}\right]$.

Now,

$|A|=3(0-5)-0+3(1+4)=-15+15=0$

$\therefore A$ is a singular matrix.

Now,

$(\operatorname{adj} A)=\left[\begin{array}{ccc}-5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6\end{array}\right]$

$\therefore(\operatorname{adj} A) B=\left[\begin{array}{lll}-5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6\end{array}\right]\left[\begin{array}{c}2 \\ -1 \\ 3\end{array}\right]=\left[\begin{array}{l}-10-10+15 \\ -6-6+9 \\ -12-12+18\end{array}\right]=\left[\begin{array}{l}-5 \\ -3 \\ -6\end{array}\right] \neq O$

Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.