Please see attached Advanced Linear Algebra question below. How to fully prove that tr(AB) = tr(BA) and tr(A) = tr(A^t)? Please do not reject question if you do not know how to solve this problem. Allow another expert a chance to answer this question. Let $ A $ and $ B $ be $ n \times n $ matrices. Recall that the trace of $ A $ is defined by\[\text{tr}(A) = \sum_{i=1}^{n} A_{ii}.\]Prove that $\text{tr}(AB) = \text{tr}(BA)$ and $\text{tr}(A) = \text{tr}(A^t)$. It's natural that we have $ \text{tr}(A) = \text{tr}(A^t) $ since $ A_{ii} = A^t_{ii} $ for all $ i $. On the other hand, we have\[\text{tr}(AB) = \sum_{i=1}^{n} (AB)_{ii} = \sum_{i=1}^{n} \sum_{k=1}^{n} A_{ik}B_{ki}\]\[= \sum_{k=1}^{n} \sum_{i=1}^{n} B_{ki}A_{ik} = \sum_{k=1}^{n} (BA)_{kk}\]\[= \text{tr}(BA).\]

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Answered: Let A and B benxn matrices. Recall that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Please see attached Advanced Linear Algebra question below.

How to fully prove that tr(AB) = tr(BA) and tr(A) = tr(A^t)?

Please do not reject question if you do not know how to solve this problem. Allow another expert a chance to answer this question.

Let $ A $ and $ B $ be $ n \times n $ matrices. Recall that the trace of $ A $ is defined by

\[
\text{tr}(A) = \sum_{i=1}^{n} A_{ii}.
\]

Prove that $\text{tr}(AB) = \text{tr}(BA)$ and $\text{tr}(A) = \text{tr}(A^t)$.

It's natural that we have $ \text{tr}(A) = \text{tr}(A^t) $ since $ A_{ii} = A^t_{ii} $ for all $ i $. On the other hand, we have

\[
\text{tr}(AB) = \sum_{i=1}^{n} (AB)_{ii} = \sum_{i=1}^{n} \sum_{k=1}^{n} A_{ik}B_{ki}
\]

\[
= \sum_{k=1}^{n} \sum_{i=1}^{n} B_{ki}A_{ik} = \sum_{k=1}^{n} (BA)_{kk}
\]

\[
= \text{tr}(BA).
\]

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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