The GATE Grind

GATE 2025 EE – Question 12

Engineering Mathematics · Linear Algebra: Eigen values, Eigen vectors · 1 mark · Multiple choice

Let $\boldsymbol v_1$ and $\boldsymbol v_2$ be the two eigenvectors corresponding to distinct eigenvalues of a $3\times3$ real symmetric matrix. Which one of the following statements is true?

  1. $\boldsymbol v_1^T\boldsymbol v_2\ne0$
  2. $\boldsymbol v_1^T\boldsymbol v_2=0$
  3. $\boldsymbol v_1+\boldsymbol v_2=\boldsymbol 0$
  4. $\boldsymbol v_1-\boldsymbol v_2=\boldsymbol 0$

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Correct answer: (B) $\boldsymbol v_1^T\boldsymbol v_2=0$

Explanation

Eigenvectors of a real symmetric matrix belonging to distinct eigenvalues are orthogonal: $\lambda_1\boldsymbol v_1^T\boldsymbol v_2=(A\boldsymbol v_1)^T\boldsymbol v_2=\boldsymbol v_1^TA\boldsymbol v_2=\lambda_2\boldsymbol v_1^T\boldsymbol v_2$, so $(\lambda_1-\lambda_2)\boldsymbol v_1^T\boldsymbol v_2=0$ and hence $\boldsymbol v_1^T\boldsymbol v_2=0$.