The values of t such that the matrix 1 3 2 2 5 t 4 7 - t - 6 has no inverse, are

Inverse of a matrix does not exist when the determinant of the matrix is zero. 

$\left|\begin{array}{ccc}1& 3& 2\\ 2& 5& t\\ 4& 7-t& -6\end{array}\right|=0$

$\Rightarrow 1\left(-30-7t+t^2\right)-3\left(-12-4t\right)+2\left(14-2t-20\right)=0$

$\Rightarrow -30-7t+t^2+36+12t-12-4t=0$

$\Rightarrow t^2+t-6=0$

$\Rightarrow t^2+3t-2t-6=0$

$\Rightarrow \left(t+3\right)\left(t-2\right)=0$

$\Rightarrow t=-3 \text{and} 2$.