Chapter 9 - Systems and Matrices - 9.3 Determinant Solution of Linear Systems - 9.3 Exercises - Page 884: 32

$$ - 1$$

$$\eqalign{ & \left| {\matrix{ 1 & 0 & 0 \cr 0 & { - 1} & 0 \cr 1 & 0 & 1 \cr } } \right| \cr & {\rm{Calculating\, the \,determinant \,by \,expanding\, the \,third\, column}} \cr & \left| {\matrix{ 1 & 0 & 0 \cr 0 & { - 1} & 0 \cr 1 & 0 & 1 \cr } } \right| = 0\left| {\matrix{ 0 & { - 1} \cr 0 & 0 \cr } } \right| - 0\left| {\matrix{ 1 & 0 \cr 0 & 0 \cr } } \right| + 1\left| {\matrix{ 1 & 0 \cr 0 & { - 1} \cr } } \right| \cr & {\rm{Solving}} \cr & \left| {\matrix{ 1 & 0 & 0 \cr 0 & { - 1} & 0 \cr 1 & 0 & 1 \cr } } \right| = 0 - 0 + 1\left( { - 1 - 0} \right) \cr & {\rm{Simplifying}} \cr & \left| {\matrix{ 1 & 0 & 0 \cr 0 & { - 1} & 0 \cr 1 & 0 & 1 \cr } } \right| = - 1 \cr} $$