Determinant as a product of eigenvalues raised to their algebraic multiplicity

• They start with the characteristic equation. The factored form makes sense.
• det(A - $\lambda I$) = $C_A$, we set this equal to 0 to find the $\lambda$s.
• Now since we want the det(A) not det(A - $\lambda I$), we set $\lambda =0$.
• So det(A) is basically the characteristic equation with $\lambda =0$.
• Doing some algebra and factoring out stuff, we can see that:

$$\det A=\prod _{i=1}^k{\lambda }_i^{a_{{\lambda }_i}}$$