The subset HC M(C) defined since it is closed under addition and multiplication, is in fact a subring. (a) Find a matrix for the inverse of x writ- ing for convenience |x| := az + a + a + az. It should not look complicated. (This shows that every nonzero element of H is invertible, making H a division ring.) 1 i (b) Next, writing ( 1 0 0 0 1 -) - - ( 1 ), * - ( ). A). j= k = -1 0 0 so that all x E H can be written x = ao + ai + a2j + a3k (where ao means a), what is x in these terms? (c) Derive a multiplication table for i, j, k (e.g., what is j, i . k, etc., in terms of linear combinations of 1 I2, i, j, and k with real coefficients). For example, i = -1 and so via ao + ai you can think of C as lying inside of H. (Congratulations! It took Hamilton 10 years to produce this multiplication table, extending the complex numbers.) -