Sample Problems – How to determine the Eigenvalues of a Matrix?

Question 1: Find the eigen value of matrix [Tex]A= \begin{bmatrix}   1 & 4 \\ 3 & 2  \\ \end{bmatrix}[/Tex].

Solution:

[Tex]A-\lambda I= \begin{bmatrix}   1-\lambda & 4 \\ 3 & 2-\lambda  \\ \end{bmatrix}[/Tex]

|A – λI|= 0

(1 – λ)(2 – λ) – 12 = 0

2 – λ – 2λ + λ² – 12 = 0

λ² – 3λ – 10 = 0

λ² – 5λ + 2λ – 10 = 0

(λ + 2)(λ – 5) = 0

λ = -2, 5

Therefore, eigen value will be (-2, 5)

Question 2: Find the eigen value of matrix [Tex]A= \begin{bmatrix}   1 & 0 & 0 \\ 0 & 1 & 2  \\ 0 & 0 & 0 \end{bmatrix}[/Tex]

Solution:

[Tex]A-\lambda I= \begin{bmatrix}   1-\lambda & 0 & 0 \\ 0 & 1-\lambda & 2  \\ 0 & 0 & 0-\lambda   \\ \end{bmatrix}[/Tex]

|A – λI| = 0

(1 – λ)[(1 – λ)(0 – λ) – 2] = 0

(1 – λ)(λ² – λ – 2) = 0

-λ³ + 2λ + λ – 2 = 0

λ = 1, 0

Therefore, the eigen value will be 1, 0.

Question 3: Find the eigen value of matrix [Tex]A= \begin{bmatrix}   4 & 1  \\ 1 & 4  \\ \end{bmatrix}[/Tex]

Solution:

[Tex]A-\lambda I = \begin{bmatrix}   4-\lambda & 1  \\ 1 & 4-\lambda   \\ \end{bmatrix}[/Tex]

[(4 – λ)(4 – λ)] – 1 = 0

16 – 4λ – 4λ + λ² – 1 = 0

λ² – 8λ + 15 = 0

λ² – 3λ – 5λ + 15 = 0

λ(λ – 3) – 5(λ – 3) = 0

(λ – 5)(λ – 3) = 0

λ = 5, 3

Therefore, the eigenvalue will be 5, 3

Question 4: Find the eigen value of the given matrix [Tex]A= \begin{bmatrix}   1 & 4 & 3  \\ 0 & 3 & 8 \\   0 & 0 & 2 \end{bmatrix}[/Tex]

Solution:

As mentioned above in the properties of eigen value i.e If a square matrix A is lower/upper triangular matrix, then its eigenvalue will be the diagonal elements of the matrix.

As the given matrix A is a lower triangular matrix so, its eigenvalue will be 1, 3, 2.

Question 5: Find the eigen value of the matrix [Tex]A= \begin{bmatrix}   2 & 2  \\ 5 & -1 \\   \end{bmatrix}[/Tex]

Solution:

[Tex]A-\lambda I= \begin{bmatrix}   2-\lambda & 2  \\ 5 & -1-\lambda\\   \end{bmatrix}[/Tex]

[(2 – λ)(-1 – λ)] – 10 = 0

-2 – 2λ + λ + λ² – 10 = 0

λ² – λ – 12 = 0

λ² – 4λ + 3λ – 12 = 0

λ(λ – 4) + 3(λ – 4) = 0

(λ – 4)(λ + 3) = 0

λ = 4, -3

Therefore, the eigenvalue will be 4, -3

Question 6: Find the eigenvalue of matrix [Tex]A= \begin{bmatrix}   -1& 8  \\  0 & -1\\   \end{bmatrix}[/Tex]

Solution:

[Tex]A-\lambda I= \begin{bmatrix}   2-\lambda & 2  \\ 5 & -1-\lambda\\   \end{bmatrix}[/Tex]

|A – λI| = 0

(-1 – λ)² – 0 = 0

(λ + 1)² = 0

λ = -1

Therefore, the eigenvalue will be -1