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GATE | GATE CS 2018 | Question 13

Last Updated : 06 Oct, 2021
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Consider a matrix A = uv^T where u = \begin{pmatrix} 1\\  2 \end{pmatrix}, v = \begin{pmatrix} 1\\  1 \end{pmatrix}. Note that v^T denotes the transpose of v. The largest eigenvalue of A is ________ . Note –This was Numerical Type question. (A) 0 (B) 1 (C) 2 (D) 3

Answer: (D)

Explanation: Given, u = \begin{pmatrix}1\\ 2\end{pmatrix}, v = \begin{pmatrix}1\\ 1\end{pmatrix} Therefore, A = uv^T = \begin{pmatrix}1\\ 2\end{pmatrix} \begin{pmatrix} 1 & 1\end{pmatrix} = \begin{pmatrix}1 & 1\\ 2 & 2\end{pmatrix} Now, A – λ I = 0 \begin{pmatrix}(1-\lambda) & 1\\ 2 & (2 - \lambda)\end{pmatrix} = 0 (1 - \lambda) (2 - \lambda) - 2 = 0 \lambda^{2} - 3 \lambda = 0 \lambda = 0, 3 So maximum is 3

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