Verify that if A = aI2 + bX for scalars a and b, then A is invertible and there are scalars d and c such that A-1 = dI2 + cX. [3]
[Note: X is a matrix satisfying the condition in problem 1 above.]
Find 4 by 4 matrices U, V, and W such that U2 = V2 = W2 = - I4, UV = W, VU = - W, VW = U, WV = - U, WU =V, and UW = - V. [2]
Verify that if B = aI2 + bU + cV + dW for scalars a, b, c, and d, then B is invertible and there are scalars p, q, r, and s such that B-1 = pI2 + qU + rV + sW. [4]
[Note: U, V, and W are matrices satisfying the conditions in problem 3 above.]
Bonus. What mathematical systems are we making representations of in 1 and 3? [1]
Department of Mathematics
Trent University