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 $R$
- If 3 is an eigenvalue of $A$, then 10 is an eigenvalue of $A^2+I$.
- 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.