Chapter 1: Systems of Linear Equations and Matrices

Transcription

1 : 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? (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.

2 Elementary Linear Algebra e Anton/Rorres 4. Which of the following matrices is in reduced row echelon form? 0 (A) (B) (C) 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? A = 5 4 8, B = [ ] [ ] [ ] (A) (B) (C) 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? (A) (B) (C) Which of the following matrices is not an elementary matrix? (A) (B) (C) [ ]

3 Elementary Linear Algebra e 3 Anton/Rorres 0. For which elementary matrix E will the equation EA = B hold? A = 0 0, B = (A) 0 0 (B) 0 0 (C) 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) 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? (A) (B) (C) The matrix is: 0 0 (A) upper triangular. (B) lower triangular. (C) both (A) and (B). neither (A) nor (B). 5 3

4 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 = 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

5 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 y = 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) What is the inverse of? Given the polynomial p(x) = x 3x + and the matrix A =, compute p(a) 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

6 Elementary Linear Algebra e 6 Anton/Rorres 8. Which elementary row operation will transform the following matrix into the identity matrix? Find the 3 3 elementary matrix that adds c times row 3 to row. 0. Find the elementary matrix E that satisfies E 0 0 = Solve the following system by inverting the coefficient matrix. 7x + y = 3x + y = 5. Solve the following matrix equation for X X = Given that A = 0 and b =, solve the system A x = b Find a nonzero solution to the following equation. 3 x = 3 x Find the values of a, b, and c that make the following matrix symmetric. 3 a b 4 0 a + b c 7

7 Elementary Linear Algebra e 7 Anton/Rorres Let A = 0 0 6, B = 4 5 0, and AB = [c ij] Find the diagonal entries c, c, and c 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 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 C = ; d =

8 Elementary Linear Algebra e 8 Anton/Rorres Answers Multiple Choice Answers. (B). (B) 3. (C) (C) 6. (C) 7. (A) 8. (A) 9. (B) 0. (C). (A). (B) 3. (B) 4. (C) Free Response Answers. 3a = b. x = t + 3, x = t + 4, x 3 = t 3. no solution 4. k =

9 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 = y + 3z = 0 6x z = 0 0. A = A = B C D Add 9 times row to row 4

10 Elementary Linear Algebra e 0 Anton/Rorres 9. 0 c E = x = 9, y = x = 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 such that x = 3x. Possible solution: x =

11 Elementary Linear Algebra e Anton/Rorres 33. C 3 H 8 + 5O 4H O + 3CO x + x x