44, 47 please

4.5 Matrix Operations 357 In Exercises 42-49, the given matrix is the augmented b) In part (a), is Ar in R?, R', or R47 Is D, in R2, matrix for a system of linear equations, Give the vector R. or R4? form for the general solution. ) Form the (2 x 2) matrix with columns 42. 012 3 0 [AB], AB,], and verify that this matrix is the product AB. d) Verify that the vector Dw is the same as 43. 10-1-2 2D + 3D2 + D3 + D4- Cre 53. Determine whether the following matrix products are defined. When the product is defined, give the 44. 0 -1 0-1 size of the product. 12 0 a) AB and BA, where A is (2 x 3) and B is (3 x 4) b) AB and BA, where A is (2 x 3) and B is (2 x 4) 12, 2 15. 0 c) AB and BA, where A is (3 x 7) and B is (6 x 3) 0 d) AB and BA, where A is (2 x 3) and B is (3 x 2) e) AB and BA, where A is (3 x 3) and B is (3 x 1) () A(BC) and (AB)C, where A is (2 x 3), B is 46. (3 x 5), and C is (5 x 4) 20, 2 g) AB and BA, where A is (4 x 1) and B is (1 x 4) -2 54. What is the size of the product (AB)(CD), where 4 A is (2 x 3), B is (3 x 4), C is (4 x 4), and D is (4 x 2)? Also calculate the size of A[B(CD) ] and 48. -2 [(AB)CID. 55. If A is a matrix, what should the symbol A mean? 27, 20 2 What restrictions on A are required in order that A? be defined? 49. 56. Set 2 0 0 0 0 3, 20 50. In Exercise 40, the calculations (AB)u and A(Bu) produce the same result. Which calculation requires fewer multiplications of individual matrix entries? (For example, it takes two multiplications to get the (1, I) entry of AB.) 51. The next section will show that all the following 10, 20 calculations produce the same result: CIA(Bu)] = (CA)(Bu) = [C(AB)]u = C[(AB) u]. Convince yourself that the first expression requires where b # 0. Show that O, A, and B are solu- the fewest individual multiplications. [Hint: Form- tions to the matrix equation X2 - 2X = O. Con- ing Bu takes four multiplications, and thus A(Bu) clude that this quadratic equation has infinitely many solutions. 18, 20 takes eight multiplications, and so on.] Count the number of multiplications required for each of the 57. Two newspapers compete for subscriptions in a four preceding calculations. region with 300,000 households. Assume that no 52. Refer to the matrices and vectors in Eq. (1 1). household subscribes to both newspapers and that the following table gives the probabilities that a a) Identify the column vectors in A = [Al, A2] household will change its subscription status during and D = [D1, D2, D3. D4]. the year. 23, 20 26, 20