# Symmetric matrix and determinant

$A$ is symmetric matrix o with $f_a(x)$ =$(x-2)^3(x+4)^2$

Than I can I receive from this $|A+2I|$?

I know minimal polynomial is $(x-2)(x+4)$ and from there $A(A+2)=8$ But what now?? Why the answer is $8^5$??

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Mary, we encourage users to accept an answer when you've found it to be helpful. You can accept only one answer to each question, but you can upvote as many answers as you'd like! –  amWhy Apr 4 at 4:07
The determinant of $A+ 2I$ is the product of all the eigenvalues of $A + 2I$. As the eigenvalues of $A$ are $\{2, 2, 2 , - 4, -4\}$, the ones of $A + 2I$ are $\{4, 4, 4, -2 , -2\}$. So the answer is $4^4$.