7 28 A. (Bpts) LetA= 3 g . Construct a 2 2 matrix B such that AR is the zero matrix. Use two different nonzero columns for B. B. (10 pts) Answer True or False. You do not need to provide justifications. (a) If A and B are 3 x 3 and B = [by bz bs] then AB = [Ab; + Ab; +Ab3]. (b} The second row of AB is the second row of A, multiplied on the right by B. (c) (AB)C = (AC)B. (d) (AB)T =ATHT. (2) The transpose of the sum of matrices equals the sum of their transposes. C. (4 pts) Suppose the first two columns of B, ie, by and ba. are equal. What can you say about the columns of AB (if AB is defined)? D. (4 pts) Suppose the second column of B is all zeros. What can you say about the second column of AB? E. (5 pts) Suppose (BC)D = 0, where B and C are m x n matrices and D is invertible. Show that B = C. F. (5 pts) Suppose A and B are n x n, B is invertible, and AB is invertible. Show that A is invertible. [Hint: Let C = AB, and solve the equation for A.] G. (5 pts) Suppose P is invertible and A = PBP~!. Solve for B in terms of A. H. (5 pts) Show that if ad be # 0, the formula for A! (where A is a 2 x 2 matrix) is correct. 1 110 L. (5 pts) LclA=[D 11 I]' {a) Construct a 4 x 2 matrix ) using only 1 and 0 as entries, such that AD =I5, (b) Is it possible that CA = I for some 4 x 2 matrix C7 Why or why not? 1. (5 pts) Assume C is 6 x 6 and the equation Cx = v is consistent for every v in BS. {a) Is 1t posible that for some v, the equation Cx = v has more than one solution? (b) Why or why not? K. (5 pts) Consider an n x n matrix H. If the equation Hx = is inconsistent for some in R, {a) what can you say about the equation Hx = 07 (b) Why? L. (5 pts) Assume L is n n and the equation Ix = 0 has the trival solution. {a) Do the columns of L span B"? (b) Justify your answer. M. (5 pts) Given a matrix A B"*" with linearly independent columns, show that the columns of A span B