Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$ if $A,B,C$ matrices have the same size.
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$\begingroup$Is it already known that $\operatorname{Tr}(XY) = \operatorname{Tr}(YX)$ when $X$ and $Y$ are square matrices of the same size?
If it is, then simply set $X= AB$ and $Y = C$. It will give you $\operatorname{Tr}(ABC) = \operatorname{Tr}(CAB)$. You can get $\operatorname{Tr}(ABC) = \operatorname{Tr}(BCA)$ in a similar fashion.
$\endgroup$ 1 $\begingroup$Hint
$$tr(ABC)=\sum_i (ABC)_{ii}=(ABC=A(BC))=\sum_i\sum_j A_{ij}(BC)_{ji}= \sum_i\sum_j\sum_k A_{ij}B_{jk}C_{ki};$$
now you can exchange the order of the matrices to arrive at the thesis as each of the $A_{ij}$, $B_{jk}$ and $C_{ki}$ are scalars (considering matrices over $\mathbb R$, for example). We arrive at
$$tr(ABC)=\sum_i\sum_j\sum_k A_{ij}B_{jk}C_{ki}=\sum_i\sum_j\sum_kB_{jk}C_{ki}A_{ij}=(BCA=(BC)A)=tr(BCA), $$
and so on.
$\endgroup$ 4 $\begingroup$Hint: use the definition of trace.
$$\text{Tr}(ABC)=\sum_i\sum_j\sum_k A_{ij}B_{jk}C_{ki}.$$
$\endgroup$ $\begingroup$I will help you by proving that $\text{tr}(AB) = \text{tr}(BA)$ for a $m \times n$ matrix $A$ and a $n \times m$ matrix $B$ first. Let $C = AB$ and $D = BA$. Then using the definition of matrix multiplication we find that $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$ and $d_{ij} = \sum_{m=1}^{n} b_{im}a_{mj}$. Then,
\begin{align*} \text{tr}(C) & = \sum_{m = 1}^n c_{mm} \\ & = \sum_{m=1}^n\sum_{k=1}^na_{mk}b_{km} \\ & = \sum_{k=1}^n\sum_{m=1}^nb_{km}a_{mk} \\ & = \sum_{k=1}^nd_{kk} \\ & = \text{tr}(D). \end{align*} Now, for $ABC$ take $AB = A_1$ and $C = A_2$ and apply the idenity proven above.
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