Verify the following properties of partitioned matrices. Given a symmetric p.d. matrix A is partitioned in the form A =  A11 A12 A21 A22  , where A11 and A22 are square matrices, both A11 and A22 are p.d. and the inverse of A is given by A−1 =  A11 + A−1 11 A12B−1A21A−1 11 −A−1 11 A12B−1 −BA21A−1 11 B−1  , with B = A22 − A21A−1 11 A12. Besides, by straightforward algebra, one can verify that (x 1 x 2 ) A−1 x1 x2  = x 1A−1 11 x1 + b B−1b, where b = x2 − A21A−1 11 x1.