Answer
No inverse, $\triangle = 0$
Work Step by Step
Let's perform row operations to make this easier. $$ \begin{vmatrix} 1 & 2 & 5 \\ -2 & -3 & 2 \\ 3 & 5 & 3 \\ \end{vmatrix}$$
Multiplying the first row by 2, while dividing the determinant by 2...
$$ 1/2 \begin{vmatrix} 2 & 4 & 10 \\ -2 & -3 & 2 \\ 3 & 5 & 3 \\ \end{vmatrix}$$
Adding the first 2 rows,
$$ 1/2 \begin{vmatrix} 2 & 4 & 10 \\ 0 & 1 & 12 \\ 3 & 5 & 3 \\ \end{vmatrix}$$
Subtracting 4 times row 2 from row 1,
$$ 1/2 \begin{vmatrix} 2 & 0 & -38 \\ 0 & 1 & 12 \\ 3 & 5 & 3 \\ \end{vmatrix}$$
Subtracting 5 times row 2 from row 3,
$$ 1/2 \begin{vmatrix} 2 & 0 & -38 \\ 0 & 1 & 12 \\ 3 & 0 & -57 \\ \end{vmatrix}$$
Dividing row 1 by 2, row 3 by 3, then multiplying the whole determinant by 6,
$$ 3 \begin{vmatrix} 1 & 0 & -19 \\ 0 & 1 & 12 \\ 1 & 0 & -19 \\ \end{vmatrix}$$
Since the first and last row are equal, the determinant is zero.
