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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)

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

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