Do matrices with the same determinant have the same characteristic polynomial?

Question

If $A$, $B$ $\in M_n(\mathbb C)$, and $det(A)=det(B)$, then would they necessarily have the same characteristic polynomial?

score 1 · Answer 1 · answered Dec 05 '13 at 00:28

1

No, if $n>1$. I'm bad at $\TeX$'ing matrices so let $A=I$, $B=-I$ if $n$ is even. If $n$ is odd, then let $A$ and $B$ be diagonal, let $A$ have diagonal elements $4, 1,1,\ldots, 1$, and let $B$ have diagonal elements $2,2,1,1,1,\ldots,1$.

answered Dec 05 '13 at 00:28

Stefan Smith

8,192

score 1 · Accepted Answer · answered Dec 05 '13 at 00:29

No: Consider $\begin{pmatrix}0&0\\ 0&0\end{pmatrix}$ and $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$.

Since $\det(A)$ is the additive constant of the characteristic polynomial you've could also posed the question: “Are two polynomials identical if their additive constants are the same?”

score 1 · Answer 3 · answered Dec 05 '13 at 00:30

1

Choose $A$ to be the zero matrix, and $B$ a non-zero non-invertible matrix with a non-zero eigenvalue. Then these matrices have equal determinant but not the same characteristic polynomial.

answered Dec 05 '13 at 00:30

score 1 · Answer 4 · answered Dec 05 '13 at 00:32

1

Consider $I_2$ and $-I_2$. What are the Eigenvalues?

answered Dec 05 '13 at 00:32

LASV

3,108

Do matrices with the same determinant have the same characteristic polynomial?

4 Answers4