Please answer each question with solutions. Thank you.

1. A matrix B is said to be a square root of a matrix A if BB = A. (a) Find two square roots of A = 2 2 (b) How many different square roots can you find of A = 3 9 ? (c) Do you think that every 2 x 2 matrix has at least one square root? Explain your reasoning. 2. Let 0 be the 2 x 2 zero matrix. (a) Is there a 2 x 2 matrix A such that A * 0 and AA = 07 Justify your answer. (b) Is there a 2 x 2 matrix A such that A * 0 and AA = A? Justify your answer. 3. The accompanying table shows a record of May and June unit sales for a clothing store. Let M denote the 4 x 3 matrix of May sales and J the 4 x 3 matrix of June sales. May Sales Small Medium Large June Sales Small Medium Large Shirts 45 60 75 Shirts 30 33 40 Jeans 30 30 40 Jeans 21 23 25 Suits 12 65 45 Suits 9 12 11 Raincoats 15 40 35 Raincoats 8 10 9 (a) What does the matrix M + J represent? (b) What does the matrix M - J represent? (c) Find a column vector z for which Ma provides a list of the number of shirts, jeans, suits, and raincoats sold in May. (d) Find a row vector y for which yM provides a list of the number of small, medium, and large items sold in May. (e) Using the matrices r and y, what does yMr represent