Question

let A,B be square matrices of the same size. Prove that tr(AB)=tr(BA).....tr denotes trace of a mtrix

Answer 1

Theorem
Let A and B be n×n matrices, then Tr(A B) = Tr (B A).
Proof

\[\mathrm{Tr}(\mathbf{AB}) = \sum_{i=1}^n (\mathbf{AB})_{ii} = \sum_{i=1}^n\sum_{j=1}^n \; A_{ij}B_{ji} = \sum_{j=1}^n\sum_{i=1}^n \; B_{ji} A_{ij} = \sum_{j=1}^n (\mathbf{BA})_{jj} = \mathrm{Tr}(\mathbf{BA}) \]

Answer 2

do there exist a 3 x 3 matrices A,B such that:

AB-BA= \[\left[\begin{matrix}1 & -2 & 6 \\ 2 & 0 & -1\\ -6 & 1 & 1\end{matrix}\right]\]

Answer 3

hint: use the result from the previous question