Linear Algebra: Properties of the Determinant

Question

On a recent exam, I was given the following problem:

Suppose that $\det(A) = -3$, $\det(A + I) = 2$, and $\det(A + 2I) = 5$.

What is $\det(A^4 + 3A^3 + 2A^2)$?

I just don't see how the first sentence relates to the second. I was hoping someone could explain what property of the determinant this question is referencing (but not necessarily give the solution).

Also: Apologies for the vague question title and my inability to use Latex (feel free to edit), as well as the homework-esque nature of this question.

Answer 1

Hints:

• $A^4 + 3A^3 + 2A^2 = A^2(A^2 + 3A + 2I) = A^2(A + I)(A + 2I)$
• $\det XY = (\det X)(\det Y)$

Answer 2

$$|A^4+3A^3+2A^2|=|A^2||A^2+3A+2I|=|A|^2|A(A+I)+2(A+I)|=$$

$$=|A|^2|A+2I||A+I|=9\times 5\times 2=90$$