Consider the following two statements with respect to the matrices A_m × n, B_n × m, C_n × n and D_n × n. Statement 1: tr(AB) = tr(BA) Statement 2: tr(CD) = tr(DC) where tr() represents the trace of a matrix. Which one of the following holds?

Both Statement 1 and Statement 2 are correct.

Statement 1 is correct and Statement 2 is wrong.

Statement 1 is wrong and Statement 2 is correct.

Both Statement 1 and Statement 2 are correct.

Both Statement 1 and Statement 2 are wrong.

GATE CSE 2022 Q20: Trace of AB vs BA for rectangular and square matrices

Consider the following two statements with respect to the matrices $A_{m \times n}$ , $B_{n \times m}$ , $C_{n \times n}$ and $D_{n \times n}$ .

Statement 1: $tr(AB) = tr(BA)$ Statement 2: $tr(CD) = tr(DC)$

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

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: C (official GATE 2022 final answer key)

Concept used: The cyclic property of trace holds whenever both products are defined, even if they have different sizes.

First step: Write $tr(AB) = \sum_i \sum_j a_{ij} b_{ji}$ and compare it with $tr(BA) = \sum_j \sum_i b_{ji} a_{ij}$ .

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Source: official GATE 2022 CS question paper and final answer key, organised by IIT Kharagpur. Question paper · Answer key