subquestion b). what's the right way and what is the reason to find eigenspace without finding eigenvector?

0 O 0 0 0 6. For the symmetric matrix A = 1 0 O O 1 O OOH O (a) find the eigenvalues of A (make good choices of row or column on which to expand by cofactors to simplify computation of the eigenvalues), and (b) without finding eigenvectors, state the dimension of each eigenspace