Final answer:
To calculate eigenspaces, solve the characteristic polynomial for eigenvalues and then solve the systems (A - λI)x = 0 for each eigenvalue to find the eigenvectors.
Step-by-step explanation:
To calculate the corresponding eigenspaces, we first need to find the eigenvalues by solving the characteristic polynomial equation, which in this case is given as -λ³ + 12λ² - 21λ - 980 = 0. After finding the eigenvalues, the next step is to determine the eigenvectors. For each eigenvalue λ, we solve the system of linear equations (A - λI)x = 0, where A is the matrix in question, I is the identity matrix, and x is the eigenvector corresponding to λ.
Once the eigenvalues are known, for each eigenvalue we need to write down the system of linear equations in matrix form to calculate the eigenvectors. Then, we use methods such as Gaussian elimination or row reduction to solve these systems and find the basis for each eigenspace. These bases will be the sets of all eigenvectors associated with each eigenvalue, along with the zero vector.
The matrix A is not explicitly given in the question, which seems to be from a larger problem set. To solve the systems, additional information on the matrix A would be necessary.