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GATE 2025 MT – Question 11

Engineering Mathematics · Linear algebra and matrices · 1 mark · Multiple choice

Which one of the following matrices has eigenvalues 1 and 6?

  1. $\begin{bmatrix}5&-2\\-2&2\end{bmatrix}$
  2. $\begin{bmatrix}3&-1\\-2&2\end{bmatrix}$
  3. $\begin{bmatrix}3&-1\\-1&2\end{bmatrix}$
  4. $\begin{bmatrix}2&-1\\-1&3\end{bmatrix}$

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Show answer and explanation

Correct answer: (A) $\begin{bmatrix}5&-2\\-2&2\end{bmatrix}$

Explanation

For a 2×2 matrix, the eigenvalues sum to its trace and multiply to its determinant. Matrix A has trace 7 and determinant 10−4=6, giving characteristic polynomial λ²−7λ+6=(λ−1)(λ−6). Thus its eigenvalues are precisely 1 and 6: **A**.