Final answer:
To find the characteristic equation and eigenvalues of the given matrix, we need to compute the determinant and solve for the roots. The eigenvalues are 8 and 1. The corresponding eigenspaces have bases { [1, -2] } and { [2, 7] } respectively.
Step-by-step explanation:
The given matrix is [ [8, -4], [-2, 1] ]. To find the characteristic equation, we need to compute the determinant of the matrix. The characteristic equation is obtained by setting the determinant equal to zero and solving for the eigenvalues. The eigenvalues are the roots of the characteristic equation.
To find the eigenvalues, we have:
|λI - A| = 0
[ [λ-8, 4], [2, λ-1] ] = 0
(λ-8)(λ-1) - 2(4) = 0
λ^2 - 9λ + 4 = 0
Using the quadratic formula, we can solve for λ:
λ = (9 ± √(9^2 - 4(1)(4))) / 2
λ = (9 ± √(81-16)) / 2
λ = (9 ± 7) / 2
λ₁ = 8
λ₂ = 1
Now, let's find the eigenvectors corresponding to each eigenvalue:
For λ₁ = 8:
Substituting λ₁ into the matrix equation: [ [8-8, 4], [2, 8-1] ] * [x, y] = [0, 0]
The resulting equation is: [ [0, 4], [2, 7] ] * [x, y] = [0, 0]
Solving this system of equations, we find a basis for the eigenspace corresponding to λ₁ as { [1, -2] }.
For λ₂ = 1:
Substituting λ₂ into the matrix equation: [ [1-8, 4], [2, 1-1] ] * [x, y] = [0, 0]
The resulting equation is: [ [-7, 4], [2, 0] ] * [x, y] = [0, 0]
Solving this system of equations, we find a basis for the eigenspace corresponding to λ₂ as { [2, 7] }.