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GATE 2026 MT – Question 12

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

Select the correct pair of eigenvectors for the symmetric matrix $$A=\begin{bmatrix}1&2\\2&4\end{bmatrix}.$$

  1. $\begin{pmatrix}1\\2\end{pmatrix}$ and $\begin{pmatrix}-2\\1\end{pmatrix}$
  2. $\begin{pmatrix}1\\2\end{pmatrix}$ and $\begin{pmatrix}2\\1\end{pmatrix}$
  3. $\begin{pmatrix}-1\\-2\end{pmatrix}$ and $\begin{pmatrix}2\\1\end{pmatrix}$
  4. $\begin{pmatrix}-1\\2\end{pmatrix}$ and $\begin{pmatrix}2\\-1\end{pmatrix}$

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Correct answer: (A) $\begin{pmatrix}1\\2\end{pmatrix}$ and $\begin{pmatrix}-2\\1\end{pmatrix}$

Explanation

Multiplying gives $A(1,2)^T=(5,10)^T=5(1,2)^T$, so the first vector in A has eigenvalue 5. Also, $A(-2,1)^T=(0,0)^T=0(-2,1)^T$, so the second has eigenvalue 0.

Both are nonzero and mutually perpendicular, as expected for distinct eigenvalues of a real symmetric matrix. The correct pair is **A**.