Please answer 24, 29, 31

der/Homeworks?preview=39002061 owar... 4.2 Echelon Form and Gauss-Jordan Elimination 325 24. x1 - x+ x= 3 a) Find the maximum value of ary such that 2x + 12 - 4ry = -3 x1 2 0 and x2 2 0. 25. x1 + $2 = 2 b) Find the maximum value of 3.x1 + 312 = 6 y = 2ry - 4x2 + xy subject to.xy 2 0 and 26. X1 - 2+ x3-4 x1 20. 2x1 - 2x2 + 3x3 = 2 c) Find the minimum value of 27. x1+ xz- x- 2 y = (x1 - 1) + (x2 + 3)2 4 (x3 + 1) with no -3x1 - 3x2 + 3x3 = -6 restriction on ly or 12. [Hint: Regard y as a 28. 2x1 + 3x2 - 4x3 = 3 function of ary and set the derivative equal to O; *1 - 2x2 - 2ng =-2 then apply the second-derivative test to verify -x1 + 1612 + 2x3 = 16 that you have found a minimum.] 29. x1+12- $3= 44. Let A and / be as follows: 2x1 - 12 + 1x3 = -x1 412 -5x3 =-5 30. x1 4 X2 X5 = 1 Prove that if b - od # 0, then A is row equivalent x2 + 2xy + 14 4 3x5 = 1 to 1. 45. As in Fig. 4.4, display all the possible configurations 31 X1 4 X3 +24 - 275 =1 for a (2 x 3) matrix that is in echelon form. [Hint: 2x1 4 X2 + 313 - x4+ 85 =0 There are seven such configurations, Consider the 3x1 - x2 + 4x3 + 104 x5= 1 various positions that can be occupied by one, no, 32. Xi + x= 1 33. X1 + 12 =1 or none of the symbols.] X1 - 12 =3 *1 - 12 =3 16. Repeat Exercise 45 for a (3 x 2) matrix, for a (3 x 3) 2x1 4x2=3 2x1 + 12 = 2 matrix, and for a (3 x 4) matrix. 34. x1 + 2x2 = 1 47. Consider the matrices B and C: 35. X1 -X2 - X3 = 1 2x1 +4x2 = 2 + x3 = 2 -X1 - 2x2 = -1 *2 + 2x3 = 3 In Exercises 36-40, find all values a for which the sys- By Exercise 44, B and C are both row equivalent to tem has no solution. matrix / in Exercise 44. Determine elementary row 36. *1 4 2x2 --3 37. + 3x2 =4 operations that demonstrate that B is row equivalent to C. ax1 - 2x2 - 5 2x1 + 612 = a 48. Repeat Exercise 47 for the matrices 38. 2x1 4 4x2 = a 39. 3x1 4 ax = 3 3x1 4 612 = 5 ax1 + 3x2 = 5 40. *1 + 0x2 = 6 ax + 2ax2 = 4 49. A certain three-digit number A equals fifteen times In Exercises 41 and 42, find all values o and B where the sum of its digits. If its digits are reversed, the 0 g a