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.
Best Answer
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.