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GATE 2022 CS – Question 20

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

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?

  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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Show answer and explanation

Correct answer: (C) Both Statement 1 and Statement 2 are correct.

Explanation

$tr(AB)=\sum_{i,j}a_{ij}b_{ji}=tr(BA)$ holds for any compatible rectangular matrices. Statement 2 is the special case with square matrices, so both are correct.