Solution: In Exercise, use Eq.(2) to find the formula for

Problem 58E In Exercise, use Eq.(2) to find the formula for the sum. If available, use linear algebra software for Exercise. (Reference Equation 2)

Problem 1E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 2E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 3E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 4E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 5E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 6E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 7E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 8E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 9E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 10E Consider the matrices are a) Either state that the matrix is in echelon form or use elementary row operations to transform it to echelon form. ________________ b) If the matrix is in echelon form, transform it to reduced echelon form.

Problem 11E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 12E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 13E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 14E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 15E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 16E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 17E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 18E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 19E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 20E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 21E each of the given matrices represents the augmented matrix for a system of linear equations. In each exercise, display the solution set or state that the system is inconsistent.

Problem 22E solve the system by transforming the augmemed matrix to reduced echelon form. 2x1 – 3x2 = 5 -4x1 + 6x2 = -10

Problem 23E solve the system by transforming the augmemed matrix to reduced echelon form. x1 – 2x2 = 3 2x1 – 4x2 = 1

Problem 24E solve the system by transforming the augmemed matrix to reduced echelon form. x1 – x2 + x3 = 3 2x1 + x2 – 4x3 = -3

Problem 25E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + x2 = 2 3x1 + 3x2 = 6

Problem 26E solve the system by transforming the augmemed matrix to reduced echelon form. x1 – x2 + x3 = 4 2x1 -2 x2 + 3x3 = 2

Problem 27E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + x2 - x3 = 2 -3x1 – 3x2 + 3x3 = -6

Problem 28E solve the system by transforming the augmemed matrix to reduced echelon form. 2x1 + 3x2 – 4x3 = 3 x1 – 2x2 -2x3 = -2 –x1 + 16x2 + 2x3 = 16

Problem 29E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + x2 – x3 = 1 2x1 – x2 + 7x3 = 8 –x1 + x2 -5x3 = -5

Problem 30E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + x2 – x5 = 1 x2 + 2x3 + x4 + 3x5 = 1 x1 – x3 + x4 – x5 = 0

Problem 31E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + x3 + x4 – 2x5 = 1 2x1 + x2 + 3x3 – x4 + x5 = 0 3x1 – x2 + 4x3 + x4 + x5 = 1

Problem 34E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + 2x2 = 1 2x1 + 4x2 = 2 –x1 – 2x2 = -1

Problem 35E solve the system by transforming the augmemed matrix to reduced echelon form. x1 – x2 –x3 = 1 x1 + x3 = 2 x2 + 2x3 = 3

Problem 32E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + x2 = 1 x1– x2 = 3 2x1 + x2 = 3

Problem 36E find all values a for which the system has no solution. x1 + 2x2 = -3 ax1 – 2x2 = 5

Problem 33E solve the system by transforming the augmemed matrix to reduced echelon form. x1 + x2 = 1 x1– x2 = 3 2x1 + x2 = 2

Problem 37E find all values a for which the system has no solution. x1 + 3x2 = 4 2x1 + 6x2 = a

Problem 38E find all values a for which the system has no solution. 2x1 + 4x2 = a 3x1 + 6x2 = 5

Problem 40E find all values a for which the system has no solution.x1 + ax2 = 6 ax1 + 2ax2 = 2

Problem 39E find all values a for which the system has no solution. 3x1 + ax2 = 3 ax1 + 3x2 = 5

Problem 42E Find all values ? and ? where 0 ? ? ? 2? and 0 ? ? ? 2?. 2 cos2 ? – sin2 ? = 1 12 cos2 ? + 8 sin2 ? = 13

Problem 41E Find all values ? and ? where 0 ? ? ? 2? and 0 ? ? ? 2?. 2 cos ? + 4 sin ? = 3 3 cos ? ? 5 sin ? = -1

Problem 43E Describe the solution set of the following system in terms of x3: x1 + x2 + x3 = 3 x1 + 2x2 = 5 For x1,x2,x3 in the solution set: a) Find the maximum value of x3 such that x1 ? 0 and x2 ? 0 ________________ b) Find the maximum value of y = 2x1 – 4x2 + x3 subject to x1 ? 0 and x1 ? 0 ________________ c) Flnd the minimum value of y = (x1 ? 1)2 + (x2 + 3)2 + (x3 + 1)2 with no restriction on x1 or x2.[Hint: Regard y as a function of x3 and set the derivative equal to 0;then apply the second – derivative test to verify that you have found a minimum.]

Problem 44E Let A and I be as follows: Prove that b – cd ? 0, then A is row equivalent to I

Problem 45E Display all the possible configurations for a (2 × 3) matrix that is in echelon form [hint:there are seven such configurations. Consider the various positions that can be occupied by one two,or none of the symbols.] Figure Patterns for four of the seven possible types of (3 × 3) matrices in echelon form. Entries marked can be either zero or nonzero.

Problem 47E Consider the matrices B and C: By Exercise B and C are both equivalent to matrix I Let A and I be as follows Determine elementary row operations that demonstrate that B is row equiralent to C

Problem 46E repeat Exercise. For a (3 × 2) matrix, for a (3 × 3) matrix, and for a (3 × 4) matrix. Display all the possible configurations for a (2 × 3) matrix that is in echelon form [hint:there are seven such configurations. Consider the various positions that can be occupied by one two,or none of the symbols.] Figure Patterns for four of the seven possible types of (3 × 3) matrices in echelon form. Entries marked can be either zero or nonzero.

Problem 48E Repeat Exercise For the matrices Exercise Consider the matrices B and C: By Exercise B and C are both equivalent to matrix I Let A and I be as follows Determine elementary row operations that demonstrate that B is row equiralent to C

Problem 49E The Main three-digit number N equals fifteen times the sum of its digits. If its digits are reversed the resulting number exceeds N by 396. The one's die it is one larger than the sum of the other two Give a linear system of three equations whose three un W n s are the digits of AT. Solve the system and find N.

Problem 50E Find the equation of the parabola, y = ax2 + bx + C, that passes through the points (-1,6), (1,4), and (2,9). [Hint: For each point given a linear equation in a, b, c.]

Problem 51E Three people play a game in which there are always two winners and losser. They have the understanding that the loser gives each winner an amount equal to what the winner already has. After three games, each has lost just once and each has $24. With how much money did each begin?

Problem 52E Find three numbers whose sum is 34 when the sum of the first and second is 7, and the sum of the second and third is 22.

Problem 53E A zoo charges $6 for adults, $3 for students, and $.50 for children. One morning 79 people enter and pay a total of $207. Determine the possible numbers of adults, students, and children

Problem 55E In Exercise, use Eq.(2) to find the formula for the sum. If available, use linear algebra software for Exercise. (Reference Equation 2)

Problem 54E Find a cubic polynomial, p(x) = a+bx+cx2 =dx3, such that p(1) = 5, p1(I) = 5, p(2) = 17, and p1(2) = 21. In Exercises 55-58, use Eq. (2) to find the formula for the sum. If available, use linear algebra software for Exercises 57 and 58.

Problem 56E In Exercise, use Eq.(2) to find the formula for the sum. If available, use linear algebra software for Exercise. (Reference Equation 2)

Problem 57E In Exercise, use Eq.(2) to find the formula for the sum. If available, use linear algebra software for Exercise. (Reference Equation 2)