Answer
$$0$$
Work Step by Step
$$\eqalign{
& \left| {\matrix{
5 & { - 3} & 2 \cr
{ - 5} & 3 & { - 2} \cr
1 & 0 & 1 \cr
} } \right| \cr
& {\rm{Calculating\, the\, determinant \,by \,expanding \,the \,third \,row}} \cr
& \left| {\matrix{
5 & { - 3} & 2 \cr
{ - 5} & 3 & { - 2} \cr
1 & 0 & 1 \cr
} } \right| = 1\left| {\matrix{
{ - 3} & 2 \cr
3 & { - 2} \cr
} } \right| - 0\left| {\matrix{
5 & 2 \cr
{ - 5} & { - 2} \cr
} } \right| + 1\left| {\matrix{
5 & { - 3} \cr
{ - 5} & 3 \cr
} } \right| \cr
& {\rm{Solving}} \cr
& \left| {\matrix{
5 & { - 3} & 2 \cr
{ - 5} & 3 & { - 2} \cr
1 & 0 & 1 \cr
} } \right| = \left( {6 - 6} \right) - 0 + \left( {15 - 15} \right) \cr
& {\rm{Simplifying}} \cr
& \left| {\matrix{
5 & { - 3} & 2 \cr
{ - 5} & 3 & { - 2} \cr
1 & 0 & 1 \cr
} } \right| = 0 \cr} $$
