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Consider the following two statements with respect to the matrices $\textit{A}_{m \times n}, \textit{B}_{n \times m}, \textit{C}_{n \times n}$ and $ \textit{D}_{n \times n}.$

Statement $1: tr \text{(AB)} = tr \text{(BA)}$

Statement $2: tr \text{(CD)} = tr \text{(DC)}$

where $tr()$ represents the trace of a matrix. Which one of the following holds?

  1. Statement $1$ is correct and Statement $2$ is wrong.
  2. Statement $1$ is wrong and Statement $2$ is correct.
  3. Both Statement $1$ and Statement $2$ are correct.
  4. Both Statement $1$ and Statement $2$ are wrong.
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Link to such theorems. Might be helpful.

Trace (linear algebra) From Wikipedia, the free encyclopedia

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2 Answers

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Answer C

Trace of a matrix and sum of eigen values of matrix are same.

$AB$ and $BA$ has same nonzero eigen values (Proof below) hence their sum of eigen values i.e.  trace should also be same.

 

Proof-

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Sir, Did you explain this in GO class lectures?
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Statement -1 is evident from the definition of matrix product 

{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\sum _{i=1}^{m}\left(\mathbf {A} \mathbf {B} \right)_{ii}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\sum _{j=1}^{n}\sum _{i=1}^{m}b_{ji}a_{ij}=\sum _{j=1}^{n}\left(\mathbf {B} \mathbf {A} \right)_{jj}=\operatorname {tr} (\mathbf {B} \mathbf {A} ).}

 

Statement-2 follows from statement-1, just that now both are square matrices

 

So, both are TRUE 

option C

Answer:

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