# Effect of squaring and the identity-matrix on eigenvalues and the characteristic polynomial

I am new to Linear Algebra, and would like some feedback regarding the following question:

True or false?

Let $$A$$ be a square matrix over $$R$$

1. If 3 is an eigenvalue of $$A$$, then 10 is an eigenvalue of $$A^2+I$$.
2. If P(t) is the characteristic polynomial of $$A$$, then $$P(t^2)$$ is a characteristic polynomial of $$A^2$$.

I worked out 1. to be true, but simply by building some examples and seeing how squaring and adding $$I$$ affects the eigenvalues.

I believe 2. is false, but I did this by just trying some numbers. I would love to know if I am right, and what the theory behind it is.

Thank you!

For 1, let $$v$$ be an eigenvector relative to $$3$$, that is, $$Av=3v$$. Then $$(A^2+I)v=A(Av)+v=A(3v)+v=3Av+v=9v+v=10v$$
The characteristic polynomial of $$A$$ is $$p(t)=\det(A-tI)$$, which has degree $$n$$ (if $$A$$ is $$n\times n$$).
Therefore $$p(t^2)$$ cannot be the characteristic polynomial of $$A^2$$, because it has degree $$2n$$.