: Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7. Which system corresponds to the following augmented matrix? 6 3 9 4 0 (A) (B) (C) x + x = 3 9x + 4x = x + x + 6x 3 = 3 9x + 4x = x + x + 6x 3 + 3x 4 = 0 9x + 4x x 4 = 0 x + 9x = 0 x + 4x = 0 6x = 0 3x x = 0 3. Which of the following statements best describes the following augmented matrix? 6 5 A = 3 4 (A) A is consistent with a unique solution. (B) A is consistent with infinitely many solutions. (C) A is inconsistent. none of the above.
Elementary Linear Algebra e Anton/Rorres 4. Which of the following matrices is in reduced row echelon form? 0 (A) 0 0 0 3 0 5 (B) 0 7 5 0 0 4 0 0 3 (C) 0 0 0 4 0 5 0 3 0 0 0 5. If the matrix A is 4, B is 3 4, C is 4, D is 4 3, and E is 5, which of the following expressions is not defined? (A) A T D + CB T (B) (B + D T )A (C) CA + CB T DBAE 6. What is the second row of the product AB? 0 3 7 A = 5 4 8, B = 6 3 9 7 9 7 [ ] [ ] [ ] (A) 8 33 5 (B) 64 48 9 (C) 50 89 99 a b 7. Which of the following is the determinant of the matrix A =? c d (A) ad bc (B) bc ad (C) bc ad ad bc 8. Which of the following matrices is not invertible? 3 6 7 7 9 0 9 3 (A) (B) (C) 4 3 4 4 6 5 9. Which of the following matrices is not an elementary matrix? 0 0 0 0 0 (A) (B) (C) 0 0 5 0 0 0 0 0 0 0 [ ] 48 89 33
Elementary Linear Algebra e 3 Anton/Rorres 0. For which elementary matrix E will the equation EA = B hold? 4 6 4 6 A = 0 0, B = 0 0 0 9 0 3 0 0 0 0 0 0 0 0 (A) 0 0 (B) 0 0 (C) 0 0 0 0 0 0 0 0 0 0. Which matrix will be used as the inverted coefficient matrix when solving the following system? 3x + x = 4 (A) 5 3 5x + x = 7 (B) (C) 5 3 5 3. What value of b makes the following system consistent? 4x + x = b x + x = 0 (A) b = (B) b = 0 (C) b = b = 3. If A is a 3 3 diagonal matrix, which of the following matrices is not a possible value of A k for some integer k? 0 0 0 0 0 0 0 0 0 (A) 0 4 0 (B) 0 6 0 (C) 0 0 4 0 0 0 0 0 9 4 0 5 0 0 0 0 0 3 0 0 4. The matrix 0 7 0 is: 0 0 (A) upper triangular. (B) lower triangular. (C) both (A) and (B). neither (A) nor (B). 5 3
Elementary Linear Algebra e 4 Anton/Rorres 5. If A is a 4 5 matrix, find the domain and codomain of the transformation T A (x) = Ax. (A) Not enough information (B) Domain: R 4, Codomain: R 5 (C) Domain: R 5, Codomain: R 5 Domain: R 5, Codomain: R 4 6. Which of the following is a matrix transformation? (A) T (x, y, z) = (yx, yz ) (B) T (x, y, z, w) = (xy, yz, zw, wx) (C) T (x, y, z) = (x +, x +, x + z, y + z) T (x, y) = (4x, 5x, x, 0) Free Response Questions. Find the relationship between a and b such that the following system has infinitely many solutions. x + y = a 3x + 6y = b. Solve the following system and use parametric equations to describe the solution set. x + x + 3x 3 = x x + x 3 = 3x + x + 4x 3 = 3 3. Determine whether the following system has no solution, exactly one solution, or infinitely many solutions. x + x = x + x = 4 5 3 6 4. Find the value of k that makes the system inconsistent. 0 k 9 5. Solve the following system using Gaussian elimination. x x 5x 3 = x + x + x 3 = 3x x + x 3 = 3
Elementary Linear Algebra e 5 Anton/Rorres 6. Solve the following system for x, y, and z. = 0 x y z + + = 3 x y z 3 = 0 x z 7. The curve y = ax 3 + bx + x + c passes through the points (0, 0), (, ), and (, ). Find and solve a system of linear equations to determine the values of a, b, and c. 8. Solve the following system for x and y. 9. Given C =, find CC T. 0 x + y = 6 x y = 0. Express the following matrix equation as a system of linear equations. 7 0 x 0 0 4 3 y = 0 6 0 z 0. Find the 3 3 matrix A = [a ij ] whose entries satisfy the condition a ij = i j.. Let A and B be n n matrices. Prove that tr (c A B) = c tr (A) tr (B). 4 0 3. What is the inverse of? 9 4 4 4. Given the polynomial p(x) = x 3x + and the matrix A =, compute p(a). 6 5. Let A, B, C, and D be n n invertible matrices. Solve for A given that the following equation holds. C DA CB = BCB 6. Prove that for any m n matrices A and B, (A B) T = A T B T. 7. Use the inversion algorithm to find the inverse of the following matrix. 0 0 0 4
Elementary Linear Algebra e 6 Anton/Rorres 8. Which elementary row operation will transform the following matrix into the identity matrix? 0 0 0 0 0 0 0 0 0 0 9 0 9. Find the 3 3 elementary matrix that adds c times row 3 to row. 0. Find the elementary matrix E that satisfies 4 6 4 6 E 0 0 = 0 0 0 9 0 3. Solve the following system by inverting the coefficient matrix. 7x + y = 3x + y = 5. Solve the following matrix equation for X. 3 3 0 0 4 X = 0 0 0 5 6 0 3 0 0 3 3. Given that A = 0 and b =, solve the system A x = b. 0 0 4. Find a nonzero solution to the following equation. 3 x = 3 x 4 3 5. Find the values of a, b, and c that make the following matrix symmetric. 3 a b 4 0 a + b c 7
Elementary Linear Algebra e 7 Anton/Rorres 3 4 3 7 6 6. Let A = 0 0 6, B = 4 5 0, and AB = [c ij]. 0 0 0 Find the diagonal entries c, c, and c 33. 7. Let the entries of a matrix A = [a ij ] be defined as a ij = i i + j + g(j), where g is a function of j. If A is a symmetric matrix, what is g(j)? 8. Prove that for any square matrix A, the matrix B = (A + A T ) is symmetric. 9. Find the domain and codomain of the transformation defined by 5 7 6 0 0 30. Find the standard matrix for the operator T : R R defined by 3x + x = w 4x = w 3. Find the standard matrix for the transformation T defined by the formula x x x 3 x 4 T (x, x, x 3 ) = (x, x 3, x x, 3x + x 3 ) 3. Prove that if T A : R 3 R 3 and T A (x) = 0 for every vector x in R 3, then A is the 3 3 zero matrix. 33. Write a balanced equation for the following chemical reaction. C 3 H 8 + O H O + CO 34. Find the quadratic polynomial whose graph passes through the points ( 0, 3 ), (, 8 ), and (, 0 ). 35. Use matrix inversion to find the production vector x that meets the demand d for the consumption matrix C. 0. 0.3 0. 8 C = 0.5 0. 0. ; d = 40 0. 0.4 0.3 6
Elementary Linear Algebra e 8 Anton/Rorres Answers Multiple Choice Answers. (B). (B) 3. (C) 4. 5. (C) 6. (C) 7. (A) 8. (A) 9. (B) 0. (C). (A). (B) 3. (B) 4. (C) 5. 6. Free Response Answers. 3a = b. x = t + 3, x = t + 4, x 3 = t 3. no solution 4. k =
Elementary Linear Algebra e 9 Anton/Rorres 5. x = 5, x =, x 3 = 6. x =, y =, z = 3 c = 0 7. System: a + b + c = 0 a + b + c = Solution: a =, b =, c = 0 8. x = ±, y = ± [ ] 9. CC T = 4 x + 7y = 0 0. 4y + 3z = 0 6x z = 0 0. A = 3 8 7 6 3. 4. 0 4 9 8 9 8 3 5. A = B C D 7. 4 0 4 0 0 4 8. Add 9 times row to row 4
Elementary Linear Algebra e 0 Anton/Rorres 9. 0 c 0 0 0 0 0 0 0. E = 0 0 0. x = 9, y = 3. 33 43 67 3 8 36 56 9 7 9 4 5 3 3. x = 9 4 4. Any x = [ x x ] 5. a = 4, b = 0, c = 4 6. c = 0, c = 0, and c 33 = 4 7. g(j) = j j 9. Domain: R 4, Codomain: R 30. 3 0 4 such that x = 3x. Possible solution: x = 3 3. 0 0 0 0 0 0 3
Elementary Linear Algebra e Anton/Rorres 33. C 3 H 8 + 5O 4H O + 3CO 34. 3 4x + x 9.85 35. x 5.50 35.0