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GATE 2023 CE (CE1) – Question 47

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple select

For the matrix $[A] = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{bmatrix}$ which of the following statements is/are TRUE?

  1. The eigenvalues of $[A]^T$ are same as the eigenvalues of $[A]$
  2. The eigenvalues of $[A]^{-1}$ are the reciprocals of the eigenvalues of $[A]$
  3. The eigenvectors of $[A]^T$ are same as the eigenvectors of $[A]$
  4. The eigenvectors of $[A]^{-1}$ are same as the eigenvectors of $[A]$

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Show answer and explanation

Correct answer: (A) The eigenvalues of $[A]^T$ are same as the eigenvalues of $[A]$; (B) The eigenvalues of $[A]^{-1}$ are the reciprocals of the eigenvalues of $[A]$; (D) The eigenvectors of $[A]^{-1}$ are same as the eigenvectors of $[A]$

Explanation

A matrix and its transpose have the same characteristic polynomial, so the same eigenvalues (A). If $Av = \lambda v$ then $A^{-1}v = \frac{1}{\lambda}v$, so the inverse has the reciprocal eigenvalues (B) and the same eigenvectors (D). The eigenvectors of $A^T$ are the left eigenvectors of $A$, which differ from those of $A$ unless $A$ is normal, and this matrix is not symmetric, so C is not true.