Characteristic polynomial of matrix with real coefficients

Question

I am reading a lecture note about matrix.

Let $p_A(x)$ be the characteristic polynomial of matrix $A$.

Can we say if $p_A(x) = p_A(-x)$, then $p_A(x)$ is a real coefficient polynomial?

Answer 1

Note that your title is ambiguous. I have taken it to mean that the matrix has real entries....

It is restrictive, but perhaps not as much as you think. In any case, a real matrix has a characteristic polynomial with real coefficients. Now, the characteristic polynomial of $$ \left( \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -4 & 0 & 0 & 0 \\ \end{array} \right) $$ is $$ x^4 + 4 $$ while the roots are $$ 1+i, 1-i, -1+i, -1-i $$