Section 2.6

• 10. a) 3x5. b) not defined. c) 3x4. d) not defined. e) not defined. f) 4x5.
• 12. Rewrite using summations and products of matrix elements, apply distributive law to those expressions, then translate back to matrix notation.
• 14. cii = aii bii.
• 18. Multiply first time second to get I. Multiply second times first to get I.
• 20. I will write a matrix as [ first row ; second row ; third row ; etc] a) [ -3/5 2/5; 1/5 1/5]. b) [1 18; 9 37]. c) [-37/125 18/125; 9/125 -1/125]. d) The inverse of part b gives part c.
• 22. (uses Exercise 17b and 16) Let At mean the transpose of A. (AAt)t = ((At)t)At = AAt. So AAt equals its transpose, so it is symmetric.
• 24. a) (using Exercise 23) A1(A2A3) takes 60,000 multiplications. (A1A2)A3 takes 18,000. So do it the second way. b) A1(A2A3) takes 300 multiplications. (A1A2)A3 takes 1,000. So do it the first way.
• 26. a) by definition of matrix multiplication. b) A^-1 A X = A^-1 B. IX = A^-1 B. X = A^-1 B.
• 28. a) [1 1; 1 1] b) [0 1; 0 0] c) [1 1; 1 0]
• 30. [ 1 0 ; 1 1; 1 1].
• 34. a) (A V B) V C = [(aij V bij) V cij] = [aij V (bij V cij)] = A V ( B V C). b) Same as a), with ^ replacing V.