Prove that $A$, a matrix of rank $3$, can't have characteristic polynomial of $p(x) = x^7 - x^5 + x^3$

Prove that $A$, a matrix of rank $3$, can't have characteristic polynomial of $p(x) = x^7 - x^5 + x^3$

My attempt to contradict:

Because of that characteristic polynomial, the matrix must be a $7 \times 7$ matrix. Also, $-\mathrm{tr}(A) = 0$, because $x^6 = 0$. $A$ is a matrix with rank of $3$ so the determinant is $0$ and therefore it isn't invertible.

I am unable to progress from this point onwards.