GATE | GATE CS 2018 | Question 13

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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