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1- Prove that if $A$ and $B$ are $n \times n$ matrices, then $\text {tr}(A B)=\text {tr}(B A) ?$

$A B=C=\left(C_{i j}\right)=\sum_{k=1}^{n} a_{i k} b_{k j}$

$B A=D=\left(D_{ij}\right)=\sum_{k=1}^{n} b_{i k} a_{k j}$

$\text {tr}(A B)=\text {tr}(c)=\sum_{i=1}^{n} c_{i i}$

$= \sum_{i=1}^{n} \sum_{k=1}^{n} a_{i k} b_{k i}=\sum_{i=1}^{n} \sum_{k=1}^{n} b_{k i} a_{i k}$

$=\sum_{k=1}^{n} \sum_{i=1}^{n} b_{k i} a_{i k}$

$=\sum_{k=1}^{n} D_{k k}=tr(D)=tr(B A)$

$tr(B A)=tr(D)=\sum_{i=1}^{n} D_{i i}$

$=\sum_{i=1}^{n} \sum_{k=1}^{n} b_{i k} a_{k i}$

$tr (A B)=tr (BA)$

2- If $A$ is any matrix, what is $t r\left(A A^{T}\right)$ equal to?

$A^{T}=B \Rightarrow b_{k j}=a_{j k}$

$A A^{T}=A B=C=\left(C_{i j}\right)=\sum_{k=1}^{n} a_{i k} b_{k j}$

$=\sum_{k=1}^{n} a_{i k} a_{j k}$

$tr(c)=\sum_{i=1}^{n} c_{i i}=\sum_{i=1}^{n} \sum_{k=1}^{n} a_{i k} a_{i k}$

$=\sum_{i=1}^{n} \sum_{k=1}^{n}\left(a_{i k}\right)^{2}$

3- Show that there is no $2 \times 2$ matrices $A$ and $B$ such that $A B-B A=I_{2}$

$I_{2}=\left(\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right)$

$tr(I_{2})=1+1=2 \Rightarrow (1)$

$tr(A B - B A)=tr(A B)-tr(B A) \Rightarrow tr(A B)=tr(B A)$

$\Rightarrow tr(A B)-tr(B A)=0 \Rightarrow (2)$

From (1) and (2)

$tr(A B-B A) \neq tr\left(I_{2}\right) \Rightarrow A B-B A \neq I_{2}$