Given matrix $A=\left(\begin{array}{ccc}1& -2& 1\\ 1& 1& -1\\ 3& 6& -5\end{array}\right)$ and $\text{adj }A=\left(\begin{array}{ccc}1& -4& 1\\ 2& 8& 2\\ 3& -12& 3\end{array}\right)$, the system of equations:$x-2y+z=5$$x+y-z=6$$3x+6y-5z=7$ has:

The system can be written as $AX=B$ where $B=[5,6,7{]}^T$.Determinant $|A|=1(-5+6)-(-2)(-5+3)+1(6-3)=1(1)+2(-2)+1(3)=1-4+3=0$.Since $|A|=0$, the system has either no solution or infinitely many solutions.We check $(\text{adj }A)B$:$(\text{adj }A)B=\left(\begin{array}{ccc}1& -4& 1\\ 2& 8& 2\\ 3& -12& 3\end{array}\right)\left(\begin{array}{c}5\\ 6\\ 7\end{array}\right)=\left(\begin{array}{c}5-24+7\\ 10+48+14\\ 15-72+21\end{array}\right)=\left(\begin{array}{c}-12\\ 72\\ -36\end{array}\right)\ne O$.Since $(\text{adj }A)B\ne O$ and $|A|=0$, the system is inconsistent and has no solution.