Chapter 11 - Systems of Equations and Inequalities - 11.3 Systems of Linear Equations: Determinants - 11.3 Assess Your Understanding - Page 742: 13

$-26$

We know that for a matrix \[ \left[\begin{array}{rrr} a & b & c \\ d &e & f \\ g &h & i \\ \end{array} \right] \] the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$ Hence here $D=4((-1)\cdot4-0\cdot(-3))-(-1)(6\cdot4-0\cdot1)+(2)(6\cdot(-3)-(-1)\cdot1)=4(-4)-(-1)(24)+(2)(-17)=-16+24-34=-26.$