(a) The MATLAB commands
A = floor(10 * rand(6)), B = A€² * A
will result in a symmetric matrix with integer entries. Why? Explain. Compute B in this way and verify these claims. Next, partition B into four 3 × 3 submatrices. To determine the submatrices in MATLAB, set
B11 = B(l : 3, 1 : 3), B12 = B(l : 3, 4 : 6)
and define B21 and B22 in a similar manner using rows 4 through 6 of B.
(b) Set C = inv(B11). It should be the case that CT = C and B21T = B12. Why? Explain. Use the MATLAB operation €² to compute the transposes and verify these claims. Next, set
E = B21 * C and F = B22 - B21 * C * B21€²
and use the MATLAB functions eye and zeros to construct

Compute H = L * D * L€² and compare it to B by computing H - B. Prove that if all computations had been done in exact arithmetic LDLT would equal B exactly.

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