Linear algebra linear systems & matrices

Question

Let A = [1 -1 0 0 1 1 1 0 -1] Find all symmetric matrices B such that AB is a symmetric matrix ?

Answer 1

Let B = [a b c; b d e; c e f] which has 6 unknowns. Then AB = [a–b, b–d, c–e; b+c, d+e, e+f; a–c, b–e, c–f], and BA⸆ = [...] so AB = BA⸆ provides 9 equations, some of which will be redundant. These are enough to solve for some of a, b, c, d, e, f in terms of others. That is, B will not be unique.