Final answer:
To find the eigenvalues of the given matrix, we solve the characteristic equation det(A - λI) = 0 by expanding the determinant and simplifying. The resulting eigenvalues are λ₁ = -3 and λ₂ = -9. To find the unit eigenvectors corresponding to each eigenvalue, we solve the equation (A - λI)x = 0 and find the solutions for x.
Step-by-step explanation:
We start by finding the eigenvalues λ. To do this, we solve the characteristic equation det(A - λI) = 0, where A is the given matrix and I is the identity matrix.
For the matrix A = [[1, 5], [-8, -11]], the characteristic equation is:
det([[1, 5], [-8, -11]] - λ[[1, 0], [0, 1]]) = 0
By expanding the determinant and simplifying the equation, we get:
λ² + 10λ + 27 = 0
Solving this quadratic equation, we get the eigenvalues λ₁ = -3 and λ₂ = -9.
To find the unit eigenvectors corresponding to each eigenvalue, we solve the equation (A - λI)x = 0, where x is a column vector.
For eigenvalue λ₁ = -3:
Solving (A - (-3)I)x = 0, we get:
[4, 5]x = 0
The solution to this system of linear equations is x = [-5, 4].
For eigenvalue λ₂ = -9:
Solving (A - (-9)I)x = 0, we get:
[10, 5]x = 0
The solution to this system of linear equations is x = [-1, 2].
Therefore, the eigenvalues of matrix A are λ₁ = -3 and λ₂ = -9, and the corresponding unit eigenvectors are [-5, 4] and [-1, 2].