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GATE 2022 BT – Question 39

Engineering Mathematics · Linear Algebra: Matrices and determinants, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple choice

A $2\times2$ matrix P has eigenvalue 2 with eigenvector $(1,0)^T$ and eigenvalue 5 with eigenvector $(1,1)^T$. What is P?

  1. $\begin{pmatrix}2&0\\0&5\end{pmatrix}$
  2. $\begin{pmatrix}2&3\\0&5\end{pmatrix}$
  3. $\begin{pmatrix}1&1\\0&1\end{pmatrix}$
  4. $\begin{pmatrix}1&1\\1&0\end{pmatrix}$

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Correct answer: (B) $\begin{pmatrix}2&3\\0&5\end{pmatrix}$

Explanation

If $P\mathbf v=\lambda\mathbf v$, then each eigenpair gives one equation. Write $P=\begin{pmatrix}a&b\\c&d\end{pmatrix}$.

**Eigenvalue 2 with eigenvector $(1,0)^T$:**
$$P\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}a\\c\end{pmatrix}=2\begin{pmatrix}1\\0\end{pmatrix}\;\Rightarrow\;a=2,\ c=0 .$$

**Eigenvalue 5 with eigenvector $(1,1)^T$:**
$$P\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}a+b\\c+d\end{pmatrix}=\begin{pmatrix}5\\5\end{pmatrix}\;\Rightarrow\;b=3,\ d=5 .$$

$$P=\begin{pmatrix}2&3\\0&5\end{pmatrix}\quad(\text{option B}).$$

Check: the trace is $7=2+5$ and the determinant is $10=2\times5$. ✓