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GATE 2025 CE (CE1) – Question 11

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple choice

Suppose $\lambda$ is an eigenvalue of matrix $A$ and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $B = A - 2I$, where $I$ is the identity matrix. Then, the eigenvalue of $B$ corresponding to the eigenvector $x$ is equal to

  1. $\lambda$
  2. $\lambda + 2$
  3. $2\lambda$
  4. $\lambda - 2$

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Correct answer: (D) $\lambda - 2$

Explanation

$Bx = (A - 2I)x = Ax - 2x = \lambda x - 2x = (\lambda - 2)x$, so the eigenvalue of $B$ for the same eigenvector is $\lambda - 2$.