(Solved): 1. (4 pts) The trace of an \( n \times n \) matrix \( C \), denoted \( \operatorname{Tr}(C) \), is ...

1. (4 pts) The trace of an \( n \times n \) matrix \( C \), denoted \( \operatorname{Tr}(C) \), is the sum of its diagonal entries. If \( A \) and \( B \) are \( 2 \times 2 \) matrices, (a) Show that \( \operatorname{Tr}(A B)=\operatorname{Tr}(B A) \). (Make sure your argument holds for any \( 2 \times 2 \) matrices \( A \) and \( B \).) (b) Find two matrices \( A \) and \( B \) such that \( A B-B A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \), or explain why no such matrices exist.