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Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$ if $A,B,C$ matrices have the same size.

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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.

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  • $\begingroup$ Thank's Dan. A proof is givnig above by A. Molendijk that $\text{tr}(AB) = \text{tr}(BA)$ for a $m \times n$ matrix $A$ and a $n \times m$ matrix $B$ $\endgroup$ – Mohamed Nov 29 '16 at 21:14
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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.

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  • $\begingroup$ here I used the associativity of the product of matrices. $\endgroup$ – Avitus Nov 10 '13 at 11:20
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    $\begingroup$ got it thanks :) $\endgroup$ – Wanderer Nov 10 '13 at 11:57
  • $\begingroup$ you are welcome! Feel free to upvote it, if it was helpful. $\endgroup$ – Avitus Nov 10 '13 at 12:15
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Hint: use the definition of trace.

$$\text{Tr}(ABC)=\sum_i\sum_j\sum_k A_{ij}B_{jk}C_{ki}.$$

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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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