## Differential Equations and Linear Algebra (4th Edition)

$abc-a^3-b^3-c^3$
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 $D=a(cb-a^2)-b(b^2-ac)+c(ba-c^2)=abc-a^3-b^3-abc+abc-c^3=abc-a^3-b^3-c^3$