The GATE Grind

GATE 2023 EC – Question 38

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Let $\boldsymbol x$ be an $n\times1$ real column vector with length $l=\sqrt{\boldsymbol x^T\boldsymbol x}$. The trace of the matrix $P=\boldsymbol x\boldsymbol x^T$ is

  1. $l^2$
  2. $\dfrac{l^2}{4}$
  3. $l$
  4. $\dfrac{l^2}{2}$

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Correct answer: (A) $l^2$

Explanation

$\mathrm{tr}(\boldsymbol x\boldsymbol x^T)=\sum_ix_i^2=\boldsymbol x^T\boldsymbol x=l^2$.