Math, asked by nayanasnayanagowda, 9 days ago

Find Eigent values & Eigen vectors of a function 5
4
1
2

Answers

Answered by marinateadrien

Answer:

hlo

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Answered by Afreenakbar

Answer:

The eigenvalues are λ1 = 1 and λ2 = 6.

The eigenvector are λ1 = 1 and λ2 = 6.

Step-by-step explanation:

To find the eigenvalues and eigenvectors of the matrix A = [[5, 4], [1, 2]], we need to solve the characteristic equation:

|A - λI| = 0

where I is the 2x2 identity matrix and λ is the eigenvalue we are trying to find.

So, we have:

|5-λ 4 | |5-λ 4 |

| | = | | = (5-λ)(2-λ) - 4 = λ² - 7λ + 6 = (λ-1)(λ-6)

| 1 2-λ| |1 2-λ|

Therefore, the eigenvalues are λ1 = 1 and λ2 = 6.

To find the eigenvectors, we need to solve the equation:

(A - λI) v = 0

where v is the eigenvector corresponding to eigenvalue λ.

For λ1 = 1, we have:

(5-1)v1 + 4v2 = 0

1v1 + (2-1)v2 = 0

which simplifies to:

4v1 + 4v2 = 0

1v1 + 1v2 = 0

The second equation is redundant, so we can ignore it. The first equation gives us v1 = -v2.

So, any vector of the form v = [t, -t] is an eigenvector corresponding to λ1 = 1.

For λ2 = 6, we have:

(5-6)v1 + 4v2 = 0

1v1 + (2-6)v2 = 0

which simplifies to:

-1v1 + 4v2 = 0

1v1 - 4v2 = 0

This system of equations has a non-trivial solution, namely v = [4, 1].

Therefore, the eigenvectors of A are:

v1 = [t, -t] for λ1 = 1

v2 = [4, 1] for λ2 = 6