Problem 3. (10 points) It is known that partial pivoting destroys symmetry. To preserve symme- try, the symmetric pivoting strategy can be used, which involves performing both row and column interchanges. The resulting factorization takes the form PAP" = LL, where P is a permutation matrix and L is lower triangular. Note that symmetric pivoting does not move off-diagonal entries to the diagonal. The diagonal of PAP" is simply a reordering of the diagonal of A. Consider the symmetric matrix 1 2 A = 1 3 1 2 1. Use symmetric pivoting described above to compute the factorization PAPT = LET. LOLT2. Use this factorization to solve the linear system Ax = b where bT = [2, 5,5]T. (Hint: PTP = I.) 3. Explain how the symmetric pivoting strategy can become unstable when all the diagonal entries of a matrix are small, particularly when calculations are performed in floating-point arithmetic