<s>[INST] Determinant of matrix $A^3 + 2A^2 - A - 5I$ Given the eigenvalues of A So A is a 3 by 3 matrix with eigenvalues -1, 1, 2. And I have to find the determinant of $$A^3 + 2A^2 - A - 5I$$
Let $u$ be the eigenvector for the eigenvalue -1. Let
$S = A^3 + 2A^2 - A - 5I$ then
$Su = \lambda u$.
$=(A^3 + 2A^2 - A - 5I)u\\
=A^3u + 2A^2u - Au - 5u$
Where do I go from here?
 [/INST] As $A$ has eigen values $-1,1,2$ you can actually see $A$ as $\begin{pmatrix}-1&0&0\\ 0&1&0\\ 0&0&2\end{pmatrix}$
If this is the case, what would be $A^3$??  what would be $A^2$?? 
$A^3=\begin{pmatrix}-1&0&0\\ 0&1&0\\ 0&0&8\end{pmatrix}$ and $A^2=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&4\end{pmatrix}$
So what would be $A^3+2A^2-A-5I$??
what would be determinant of $A^3+2A^2-A-5I$??
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