# If two matrices have the same trace and determinant, do they have the same characteristic polynomial?

"If two matrices have the same trace and determinant, do they have the same characteristic polynomial?"

This is more of a number theory problem if we use the eigenvalue definition for trace and determinant and I have never studied any number theory before. If I gave you n numbers and told you the sum equals p and product equals q, and product equals p, and gave you another set of n numbers also with the same sum and product, are these sets the same set of numbers?

• if the matrices are size 2 by 2, the characteristic polynomial really is $x^2 - Tx + D$ where $T$ is the trace and $D$ is the determinant. For 3 by 3 or bigger, the coefficient of $x$ is not determined by those – Will Jagy Mar 27 at 2:13
• Concise perfect answer @WillJagy. – A rural reader Mar 27 at 2:28

Hint. Consider the $$n\times n$$ zero matrix and any $$n\times n$$ singular matrix whose trace is zero.
It follows that when $$n\ge3$$, there always exist two $$n\times n$$ matrices with the same traces and determinants but different characteristic polynomials: just pick two monic degree-$$n$$ polynomials with equal coefficients of $$x^{n-1}$$ and equal constant terms, but different coefficients of $$x^k$$ for some $$0. Then their companion matrices will have equal traces and equal determinants, but different characteristic polynomials by construction.
When $$n=2$$, the characteristic polynomial of a matrix $$A$$ is $$x^2-\operatorname{tr}(A)x+\det(A)$$. Hence it is uniquely determined by the trace and determinant. The same conclusion holds when $$n=1$$, provided that the given trace is equal to the given determinant.