GATE | GATE CS 2008 | Question 28

How many of the following matrices have an eigenvalue 1?

(A) one
(B) two
(C) three
(D) four

Answer: (A)

Explanation: EigenValues of $\begin{pmatrix} 1 &0 \\ 0 &0 \end{pmatrix}$ is given by $\begin{pmatrix}1-\lambda &0 \\ 0 &0-\lambda \end{pmatrix}= 0 \Rightarrow -\lambda(1 - \lambda) = 0 \Rightarrow \lambda=1$ or $\lambda=0$

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EigenValues of $\begin{pmatrix}0 &1 \\ 0 &0 \end{pmatrix}$ is given by $\begin{pmatrix}0-\lambda &1 \\ 0 &0-\lambda \end{pmatrix}= 0 \Rightarrow \lambda^2 = 0 \Rightarrow \lambda=0$ or $\lambda=0$

EigenValues of $\begin{pmatrix}1 &-1 \\ 1 &1 \end{pmatrix}$ is given by

$\begin{pmatrix}1-\lambda &-1 \\ 1 &1-\lambda \end{pmatrix}= 0 \Rightarrow \lambda^2 - 2\lambda + 2 = 0 \Rightarrow \lambda= \frac{2\pm \sqrt{4-8}}{2}$
EigenValues of $\begin{pmatrix}-1 &0 \\ 1 &-1 \end{pmatrix}$ is given by $\begin{pmatrix}-1-\lambda &0 \\ 1 &-1-\lambda \end{pmatrix}= (\lambda+1)^2 = 0 \Rightarrow \lambda=-1$

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