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GATE 2022 EE – Question 40

Engineering Mathematics · Linear Algebra: Eigen values, Eigen vectors · 2 marks · Multiple choice

$e^A$ denotes the exponential of a square matrix A. Suppose $\lambda$ is an eigenvalue and $v$ is the corresponding eigen-vector of matrix A.

Consider the following two statements:

Statement 1: $e^\lambda$ is an eigenvalue of $e^A$.

Statement 2: $v$ is an eigen-vector of $e^A$.

Which one of the following options is correct?

  1. Statement 1 is true and statement 2 is false.
  2. Statement 1 is false and statement 2 is true.
  3. Both the statements are correct.
  4. Both the statements are false.

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Correct answer: (C) Both the statements are correct.

Explanation

Since $Av=\lambda v$, we get $A^kv=\lambda^kv$, so $e^Av=\sum\frac{A^k}{k!}v=\sum\frac{\lambda^k}{k!}v=e^\lambda v$. So $e^\lambda$ is an eigenvalue of $e^A$ with the same eigenvector $v$.