(a) Find the eigenvalues, eigenvectors and eigenspaces forA =1 0 53 6 02 0 10(b) Calculate the determinant and trace of A directly and using eigen-values.

Sure, let's solve this step by step.

(a) To find the eigenvalues of the matrix A, we need to solve the characteristic equation, which is given by det(A - λI) = 0, where I is the identity matrix and λ are the eigenvalues.

The matrix A - λI is:

A - λI = [1-λ 0 5]
         [3 6-λ 0]
         [2 0 1-λ]

So, det(A - λI) = (1-λ)((6-λ)(1-λ)) - 0 - 530 = λ^3 - 8λ^2 + 7λ = λ*(λ-1)*(λ-7) = 0

Setting this equal to zero gives the roots λ1=0, λ2=1, λ3=7. These are the eigenvalues of A.

Next, we find the eigenvectors by plugging each eigenvalue back into the equation (A - λI)v = 0 and solving for v.

For λ1=0, we have:

[1 0 5] [v1]   [0]
[3 6 0] [v2] = [0]
[2 0 1] [v3]   [0]

Solving this system gives the eigenvector v1 = (0, 0, 0).

For λ2=1, we have:

[0 0 5] [v1]   [0]
[3 5 0] [v2] = [0]
[2 0 0] [v3]   [0]

Solving this system gives the eigenvector v2 = (1, -1, 0).

For λ3=7, we have:

[-6 0 5] [v1]   [0]
[3 -1 0] [v2] = [0]
[2 0 -6] [v3]   [0]

Solving this system gives the eigenvector v3 = (1, 1, 1).

The eigenspaces are the sets of all eigenvectors corresponding to each eigenvalue, including the zero vector. So the eigenspaces are E0 = {(0, 0, 0)}, E1 = {(1, -1, 0)}, and E7 = {(1, 1, 1)}.

(b) The determinant of A is the product of its eigenvalues, so det(A) = 017 = 0.

The trace of A is the sum of its eigenvalues, so tr(A) = 0 + 1 + 7 = 8.

These values can also be calculated directly from the matrix A. The determinant is 161 - 0 - 532 = 0, and the trace is 1 + 6 + 1 = 8. So the results are consistent.