Precalculus (10th Edition)

$x=0$ or $x=-9$.
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 $D=x(x\cdot(-2)-0\cdot1)-2(1\cdot(-2)-0\cdot6)+3(1\cdot1-x\cdot6)=0\\x(-2x)-2(-2)+3(1-6x)=7\\-2x^2+4+3-18x=7\\2x^2+18x=0\\2x(x+9)=0.$ Hence by the zero product property $x=0$ or $x+9=0\\x=-9$.