Problem Consider the symmetric 6 × 6 matrix over R 1 1 1 1 1 -1 -1 1 -1 1 A = 1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 -1 1 This matrix plays a role in the construction of the icosahedron which is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. (i) Find the eigenvalues of this matrix. (ii) Consider the matrix A+ V5I6. Find the eigenvalues. (iii) The matrix A+ V5I6 induces an Euclidean structure on the quotient space R°/ker(A + V516). Find the dimension of ker(A + V5I6). 11L

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