Answer
$$186$$
Work Step by Step
$$\eqalign{
& \left| {\matrix{
4 & { - 7} & 8 \cr
2 & 1 & 3 \cr
{ - 6} & 3 & 0 \cr
} } \right| \cr
& {\rm{Calculating\, the\, determinant\, by \,expanding\, the\, second\, row}} \cr
& \left| {\matrix{
4 & { - 7} & 8 \cr
2 & 1 & 3 \cr
{ - 6} & 3 & 0 \cr
} } \right| = - 2\left| {\matrix{
{ - 7} & 8 \cr
3 & 0 \cr
} } \right| + 1\left| {\matrix{
4 & 8 \cr
{ - 6} & 0 \cr
} } \right| - 3\left| {\matrix{
4 & { - 7} \cr
{ - 6} & 3 \cr
} } \right| \cr
& {\rm{Solving}} \cr
& \left| {\matrix{
4 & { - 7} & 8 \cr
2 & 1 & 3 \cr
{ - 6} & 3 & 0 \cr
} } \right| = - 2\left( {0 - 24} \right) + \left( {0 + 48} \right) - 3\left( {12 - 42} \right) \cr
& {\rm{Simplifying}} \cr
& \left| {\matrix{
4 & { - 7} & 8 \cr
2 & 1 & 3 \cr
{ - 6} & 3 & 0 \cr
} } \right| = 48 + 48 + 90 \cr
& \left| {\matrix{
4 & { - 7} & 8 \cr
2 & 1 & 3 \cr
{ - 6} & 3 & 0 \cr
} } \right| = 186 \cr} $$
