a) find elementary matrices E1 , E2 , and E3 such that there holds: E1 A = B, E2 B = C
and E3 B = D.
b) Does there exist an elementary matrix E4 such that E4 A = C? Explain your answer. Question 2 (25 points) Given the matrix
2 2 3 F = 3 1 3 , 1 0 1 a) compute F −1 , or explain why F is not invertible. b) Determine whether the matrices F 2 , F T are invertible. Justify your answer. Note: you do not need to calculate the inverses, if they exist. Your answer should not require lengthy calculations. Question 3 (20 points) Find all solutions of the system 2x + 2y + 3z = 1 3x + y + 3z = 1 x + z = 1. Question 4 (20 points) Give conditions on b1 , b2 and b3 such that the following system is consistent: x1 − 2x2 + 5x3 = b1 4x1 − 5x2 + 8x3 = b2 −3x1 + 3x2 − 3x3 = b3 . Question 5 (15 points) Suppose that an (n × n)–matrix G satisfies G2 − 8G + 2I = 0, where I denotes the (n × n)–identity matrix and 0 denotes the (n × n)–zero matrix. Show that G is invertible, and satisfies: G−1 = 21 (8I − G).
