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

Answer

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

 

3Comments
KEVAL @kevalmalde
27 Aug 2017 11:09 am

option 4 has only one eigan value which is -1

shivani @shivani1234
27 Aug 2017 05:56 pm

go through matrix 4 properly it has 

λ=0  and  −2

Rishi Yadav @rishiyadav
28 Aug 2017 07:32 am

only one eigen value