The GATE Grind

GATE 2022 EC – Question 37

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Let $\alpha,\beta$ be two non-zero real numbers and $v_1,v_2$ be two non-zero real vectors of size $3\times1$. Suppose that $v_1$ and $v_2$ satisfy $v_1^Tv_2=0$, $v_1^Tv_1=1$, and $v_2^Tv_2=1$. Let $A$ be the $3\times3$ matrix given by:

$$A=\alpha v_1v_1^T+\beta v_2v_2^T$$

The eigenvalues of $A$ are ________.

  1. $0,\ \alpha,\ \beta$
  2. $0,\ \alpha+\beta,\ \alpha-\beta$
  3. $0,\ \frac{\alpha+\beta}{2},\ \sqrt{\alpha\beta}$
  4. $0,\ 0,\ \sqrt{\alpha^2+\beta^2}$

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Show answer and explanation

Correct answer: (A) $0,\ \alpha,\ \beta$

Explanation

$v_1,v_2$ are orthonormal. $Av_1=\alpha v_1(v_1^Tv_1)+\beta v_2(v_2^Tv_1)=\alpha v_1$ and $Av_2=\beta v_2$, so $\alpha$ and $\beta$ are eigenvalues. Any vector $v_3$ orthogonal to both satisfies $Av_3=0$, so the third eigenvalue is $0$.