Chapter 3 Linear Systems and Matrices - 3.7 Evaluate Determinants and Apply Cramer's Rule - Guided Practice for Examples 1 and 2 - Page 204: 3

$-10$

We know that for a matrix \[ \left[\begin{array}{rrr} a & b & c \\ d &e & f \\ g &h & i \\ \end{array} \right] \] the determinant is $D=a(ei-fh)-b(di-fg)+c(dh-eg).$ Hence here $D=10(-12\cdot2-4\cdot(-7))-(-2)(2\cdot(-2)-4\cdot0)+3(2\cdot(-7)-(-12)\cdot0)=10(4)+2(-4)+3(-14)=-10$