108k views
3 votes
Find eigenvalues and eigenvectors for the matrix [[ -1 1; -6 4 ]]. The smaller eigenvalue

User Saint
by
8.5k points

1 Answer

3 votes

Final answer:

The eigenvalues of the matrix [[ -1 1; -6 4 ]] are -1 and 2. After solving the characteristic equation, we substitute the eigenvalues back into the system to find their corresponding eigenvectors, focusing on the smaller eigenvalue, -1, as requested.

Step-by-step explanation:

To find the eigenvalues and eigenvectors of the matrix [[ -1 1; -6 4 ]], we need to solve the characteristic equation: det(A - λI) = 0, where A is our matrix and λ represents the eigenvalues. Let's subtract λ times the identity matrix from A and then find the determinant of the resulting matrix:


(A - λI) =
[[ -1 1; -6 4 ]] - [[λ 0; 0 λ]] =
[[ -1-λ 1; -6 4-λ]]


det(A - λI) = (-1-λ)(4-λ) - (1)(-6) = λ^2 -3λ - 2

Solving the quadratic equation λ^2 -3λ - 2 = 0 gives us the eigenvalues λ1 = -1 and λ2 = 2. Now we find the eigenvectors by substituting each eigenvalue back into the equation (A-λI)x = 0 and solving for x.

For λ1 = -1:



The solved system for λ1 gives us the eigenvector associated with λ1 = -1. Similarly, we would find the eigenvector for λ2 = 2.

The student asked for the smaller eigenvalue, so our focus would be on the eigenvalue λ1 = -1 and its corresponding eigenvector.

User Giesburts
by
8.2k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories