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.