Answer
$$ - 71$$
Work Step by Step
$$\eqalign{
& \left| {\matrix{
7 & { - 1} & 1 \cr
1 & { - 7} & 2 \cr
{ - 2} & 1 & 1 \cr
} } \right| \cr
& {\rm{Calculating\, the\, determinant\, by \,expanding\, the\, third\, column}} \cr
& \left| {\matrix{
7 & { - 1} & 1 \cr
1 & { - 7} & 2 \cr
{ - 2} & 1 & 1 \cr
} } \right| = 1\left| {\matrix{
1 & { - 7} \cr
{ - 2} & 1 \cr
} } \right| - 2\left| {\matrix{
7 & { - 1} \cr
{ - 2} & 1 \cr
} } \right| + 1\left| {\matrix{
7 & { - 1} \cr
1 & { - 7} \cr
} } \right| \cr
& {\rm{Solving}} \cr
& \left| {\matrix{
7 & { - 1} & 1 \cr
1 & { - 7} & 2 \cr
{ - 2} & 1 & 1 \cr
} } \right| = \left( {1 - 14} \right) - 2\left( {7 - 2} \right) + 1\left( { - 49 + 1} \right) \cr
& {\rm{Simplifying}} \cr
& \left| {\matrix{
7 & { - 1} & 1 \cr
1 & { - 7} & 2 \cr
{ - 2} & 1 & 1 \cr
} } \right| = - 71 \cr} $$
