If $A$ and $B$ are $n\times n$ matrices of a field $F$, then show that $\text{trace}(AB)=\text{trace}(BA)$. Hence show that similar matrices have the same trace.
I've done the first part (proving that $AB$ and $BA$ have the same trace). I can show that here if you say so. But I'm stuck on the 'Hence show' part. Please give me some ideas.
Hint: By very definition, two matrices $A,B$ are similar iff there exists an invertible matrix $S$ such that $A=SBS^{-1}$. Now apply the trace on both sides, and conclude using associativity of the matrix product.
You can use the characteristic polynomials. If $A$ is similar to $B$, then their characteristic polynomials $f_A(x)$ and $f_B(x)$ are identical. Since $$ f_A(x)=x^n-tr(A)x^{n-1}+\ldots\pm\det(A)\\ $$ and $$ f_B(x)=x^n-tr(B)x^{n-1}+\ldots\pm\det(B) $$ it follows in particular that $tr(A)=tr(B)$.
It is very clear if you focus on the component. Let's say $$M = ABC$$then the trace in tensor notation is
$$M^i_i = A^i_jB^j_kC^k_i$$
now you can see you can interchange them to form even permutation (so its $ABC,CAB, BCA$)without changing the invariant. Of course this works for any numbers of matrix multiplication as long as you got them in the right order
In so you push the $S^{-1}$ to the front and combine it with $S$, you will get identity and solve our problem
Suppose that $A$ is simular ti $B$, they there exist $C$ such that: $B=CAC^{-1}$. So: $tr(B)=tr(CAC^{-1})$, now let me call $X=CA$, and $Y=C^{-1}$, then $tr(B)=tr(XY)=tr(YX)=tr(C^{-1}CA)=tr(A)$.
