Final answer:
The matrix has eigenvalues 3 and -2, with eigenvectors (-10, 0, 9) corresponding to 3 and (10, 0, -9) corresponding to -2, thus providing a basis for their respective eigenspaces.
Step-by-step explanation:
The question asks to find the eigenvalues of a given matrix and determine a basis for each eigenspace. The matrix provided is:
[3 0 0]
[9 3 10]
[-5 0 -2]
To find the eigenvalues, we must solve the characteristic equation det(A - λI) = 0, where A is our matrix and λ is the eigenvalue. Subtracting λ from the diagonal entries of A, we have:
det([3-λ 0 0]
[9 3-λ 10]
[-5 0 -2-λ]) = 0
The determinant of this matrix (which is a diagonal matrix) is the product of its diagonal entries: (3-λ)(3-λ)(-2-λ) = 0. This gives us the eigenvalues λ = 3, and λ = -2.
Next, we find the eigenvectors by substituting λ back into (A - λI)x = 0 and solving for the vector x. For λ = 3, we get:
[0 0 0]
[9 0 10]
[-5 0 -5]
Which simplifies to the equations 9x1 + 10x3 = 0 and -5x1 - 5x3 = 0. From this, one possible eigenvector for λ = 3 is x = (-10, 0, 9).
For λ = -2, we have:
[5 0 0]
[9 5 10]
[-5 0 0]
This leads us to the equation 9x1 + 10x3 = 0, from which an eigenvector x can be (10, 0, -9).
Therefore, a basis for the eigenspace corresponding to λ = 3 is {(-10, 0, 9)}, and for λ = -2 it is {(10, 0, -9)}.