5 -4 4 The characteristic polynomial for the matrix G 4 -5 8 2 -4 7 is Ag(t) = (t – 3)²(t – 1). Find bases for the eigenspaces Ez and E1.

a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.

Determine all nn symmetric matrices that have 0 as their only eigenvalue.