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GATE 2026 EE – Question 30

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

Two $n\times n$ matrices $A$ and $B$ have a common eigenvalue 2, and the same corresponding nonzero eigenvector.

Which of the following options is/are correct?

(Note: $I$ is the $n\times n$ identity matrix.)

  1. Determinant $(A-2I)=0$
  2. Determinant $(B-2I)=0$
  3. Determinant $(A+B-2I)=0$
  4. Determinant $(A+B-4I)=0$

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Show answer and explanation

Correct answer: (A) Determinant $(A-2I)=0$; (B) Determinant $(B-2I)=0$; (D) Determinant $(A+B-4I)=0$

Explanation

If $A\boldsymbol v=2\boldsymbol v$ and $B\boldsymbol v=2\boldsymbol v$ with $\boldsymbol v\ne0$, then $(A-2I)\boldsymbol v=0$ and $(B-2I)\boldsymbol v=0$, so both determinants are zero (A, B). $(A+B-4I)\boldsymbol v=2\boldsymbol v+2\boldsymbol v-4\boldsymbol v=0$, so that determinant is also zero (D). $(A+B-2I)\boldsymbol v=2\boldsymbol v\ne0$, so nothing forces this determinant to vanish (C).