se equations when answering the following questions: 5x + y + z = K x + 7y + z = K x + y + 8z = K a. Express the equations as a homogeneous linear system of three equations in four unknowns (x, y, z, and K), and show that the solution set has one arbitrary parameter. b. Find the smallest solution for which all four variables are positive integers. c. Show that the solution given in the example is included among your solutions. Assignment 2 This assignment, based on the content of Unit 2 of the Study Guide, is worth 3% of your grade. We recommend that you hand it in after you complete Unit 2. You must show all of your work in order to obtain full marks. For your convenience, each exercise mentions the section of the Study Guide that corresponds to the given problem. Note: Each of the questions below is of equal weight, and each will be marked out of twenty (20) points. To gain full marks, you must show all your work. 20 points 1. (Section 2.1) 2 a. Given A = 4 i. 0 1 compute the following, A3 ii. A3 iii. A2 2A + I b. If a polynomial p(x) can be factored as a product of two lower degree polynomials p1 (x) and p2 (x) as p(x) = p1 (x)p2 (x) then it can be proved that p(A) = p1 (A)p2 (A) for an arbitrary square matrix A. Verify this statement for polynomials p(x) = x2 9, 20 points Mathematics 270: Linear Algebra I 2. p1 (x) = x + 3, p2 (x) = x 3. (Section 2.2) a. Use the inversion algorithm to find the inverse of the matrix, if the inverse exists. 1 0 0 0 1 3 0 0 1 3 5 0 1 3 5 7 b. Find all values of c, if any, for which the given matrix is invertible. c 1 0 1 c 1 0 1 c c. Show that the matrices A and B are row equivalent by finding a sequence of elementary row operations that produces B from A, and then use that result to find a matrix C such that CA = B. 2 1 0 6 9 4 A = 1 1 , B = 0 5 1 0 1 2 1 3 0 1 Assignment 2 1 20 points 3. (Section 2.3) a. Solve the system by inverting the coefficient matrix and using Theorem 1.6.2 of the textbook. x 2y 3z = 0 w + x + 4y + 4z = 7 w + 3x + 7y + 9z = 4 w 2x 4y 6z = 6 b. Determine conditions on the bi 's, if any, in order to guarantee that the linear system is consistent. x1 2x2 x3 = b1 4x1 + 5x2 + 2x3 = b2 4x1 + 7x2 + 4x3 = b3 20 points 4. (Section 2.4) a. Find the diagonal entries of AB by inspection. 6 0 0 4 0 0 A = 2 0 0, B = 1 5 0 3 2 6 3 0 7 b. 20 points 5. Find a diagonal matrix A, given 9 0 0 A2 = 0 4 0 0 0 1 (Section 2.5) Consider an open economy with consumption matrix 1/2 1/4 1/4 C = 1/2 1/8 1/4 1/2 1/4 1/8 If the open sector demands the same dollar value from each product-producing sector, which such sector must produce the greatest dollar value to meet the demand? Is the economy productive? 2 Assignment 2 Mathematics 270: Linear Algebra I Refer to these equations when answering the following questions: 5x + y + z = K x + 7y + z = K x + y + 8z = K a. Express the equations as a homogeneous linear system of three equations in four unknowns (x, y, z, and K), and show that the solution set has one arbitrary parameter. b. Find the smallest solution for which all four variables are positive integers. c. Show that the solution given in the example is included among your solutions. Refer to these equations when answering the following questions: 5x + y + z = K x + 7y + z = K x + y + 8z = K a. Express the equations as a homogeneous linear system of three equations in four unknowns (x, y, z, and K), and show that the solution set has one arbitrary parameter. b. Find the smallest solution for which all four variables are positive integers. c. Show that the solution given in the example is included among your solutions. (Section 2.3) 3. a. Solve the system by inverting the meicient matrix and using 'I'heanml 1.6.2 ofle tademk. :: 2y3z =D w+m +4y+4z =7 w+3sc +7y+Qz =4 w-2$4yz=6 b. Deten'nine condionsonthabi's, ifany, inmertoguarmteematthe linear systemis cmsisteut 931 "-'-'I?2"='3 =51 -4:51 +532 +233 2 b: 421 + 7m + 433 = b; \f