##### Example on Eigen Value

How many of the following matrices have an eigenvalue 1?

$\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$$\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}$$\begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix}$ and $\begin{bmatrix} -1 & 0 \\ 1 & -1 \\ \end{bmatrix}$

a) 1

b) 2

c) 3

d) 4

For a matrix $A$, and eigen value $\lambda$,

$|A - \lambda I | = 0$ is the characteristic equation.

Lets look at the eigen values for each matrix given in the question

(i) for $\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$

$\begin{vmatrix} 1-\lambda & 0 \\ 0 & - \lambda\\ \end{vmatrix} = 0$

$(1- \lambda)(- \lambda) = 0$

⇒ $\lambda = 0 \ \ and \ \ 1$

(ii) For $\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}$ ,

$\begin{vmatrix} - \lambda & 1 \\ 0 & -\lambda \\ \end{vmatrix} = 0$

⇒ $\lambda^2 - 0 = 0$

$\lambda = 0 \ \ and \ \ 0$

(iii) for $\begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix}$

$\begin{vmatrix} 1- \lambda & -1 \\ 1 & 1-\lambda \\ \end{vmatrix} = 0$

⇒ $(1 - \lambda)^2 + 1 = 0$

$(1 - \lambda)^2 - (-1)^2 = 0$

⇒ $(1 - \lambda -1) (1 - \lambda + 1) = 0$

$(- \lambda)(2 - \lambda) = 0$

⇒ $\lambda = 0 \ \ and \ \ 2$

(iv) for $\begin{bmatrix} -1 & 0 \\ 1 & -1 \\ \end{bmatrix}$,

$\begin{vmatrix} -1 - \lambda & 0 \\ 1 & -1 - \lambda \end{vmatrix} = 0$

$(-1 - \lambda)^2 - 1^2 = 0$

⇒ $(-\lambda) (- \lambda - 2) = 0$

⇒ $\lambda = 0 \ \ and \ \ -2$

Therefore, Ans is clearly (a) 1

KEVAL
27 Aug 2017 11:09 am

option 4 has only one eigan value which is -1

shivani
27 Aug 2017 05:56 pm

go through matrix 4 properly it has

λ=0  and  −2