MATH 1B03: Midterm 1 - VERSION 1 Instructor: Adam Van Tuyl Date: October 4, :00PM Duration: 75 min.

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1 MATH B3: Midterm - VERSION Instructor: Adam Van Tuyl Date: October 4, 27 7:PM Duration: 75 min. Name: ID #: Instructions: This test paper contains 2 multiple choice questions printed on both sides of the page. The questions are on pages 2 through. Scrap paper is available for rough wor. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COM- PLETE. BRING ANY DISCREPANCIES TO THE ATTENTION OF THE INVIGILA- TOR. Select the one correct answer to each question and ENTER THAT ANSWER INTO THE SCAN CARD PROVIDED USING AN HB PENCIL. You are required to submit this boolet along with your answer sheet. HOWEVER, NO MARKS WILL BE GIVEN FOR THE WORK IN THIS BOOKLET. Only the answers on the scan card count for credit. Each question is worth mar. The test is graded out of 2. There is no penalty for incorrect answers. NO CALCULATORS ALLOWED. Computer Card Instructions: IT IS YOUR RESPONSIBILITY TO ENSURE THAT THE ANSWER SHEET IS PROPERLY COMPLETED. YOUR TEST RESULTS DEPEND UPON PROPER ATTENTION TO THESE INSTRUCTIONS. The scanner that will read the answer sheets senses areas by their non-reflection of light. A heavy mar must be made, completely filling the circular bubble, with an HB pencil. Mars made with a pen or felt-tip marer will NOT be sensed. Erasures must be thorough or the scanner may still sense a mar. Do NOT use correction fluid. Print your name, student number, course name, and the date in the space provided at the top of Side (red side) of the form. Then the sheet MUST be signed in the space mared SIGNATURE. Mar your student number in the space provided on the sheet on Side and fill the corresponding bubbles underneath. Your student number MUST be 9 digits long. If you have a student number that is 7 digits, begin your student number with (two zeroes). Mar only ONE choice (A, B, C, D, E) for each question. Begin answering questions using the first set of bubbles, mared. McMaster University Math B3 Fall 27 Page of

2 McMaster University Math B3 Fall 27 Page 2 of. Which equation is NOT linear in x, x 2 and x 3? (a) x + x 2 + x 3 = 27. (b) (ln (27))x + 27x 2 + (27!)x 3 = 7. (c) x x 2 x 3 + x 2 x 2 2x e x 3 = 4. (d) (sin (27))x + (cos (27))x 3 =. (e) x + 2x 2 + 4x =. 2. Which of the following matrices are in reduced row echelon form? i) v) ii) 5 π 6 4 iii) iv) 27 (a) iii) and v) only (b) All of them (c) None of them (d) v) and i) only (e) i) only McMaster University Math B3 Fall 27 Page 2 of

3 McMaster University Math B3 Fall 27 Page 3 of 3. Suppose that you want to find a parabola y = ax 2 + bx + c that passes through the (x, y) pairs ( 5, 7), (, ), and (, 2). To obtain the coefficients a, b, and c you would try to solve a system of linear equations whose augmented matrix is which of the following? (a) (d) (b) (e) (c) 4. The following system of linear equations has how many solutions? x 7x 2 + x 3 + 6x 4 = 5 x + x 2 + x 3 2x 4 = 3 x + 7x 2 4x 3 + 2x 4 = 7 (a) None (b) One (c) Two (d) 27 (e) Infinitely many McMaster University Math B3 Fall 27 Page 3 of

4 McMaster University Math B3 Fall 27 Page 4 of For the next two questions, use the following three matrices: A = [ ] 4 5 3, B = 5 4, C = What matrix multiplication will yield a 2 2 matrix? (a) ABC (b) C T A T BC (c) A T B T AC T (d) C T BC (e) None of the above 6. What number is the largest? (a) tr(aa T ) (b) tr(3b) (c) tr(cc T ) 7. If A, B, C and D are invertible matrices of the same size and (2BA C) = D which of the following must be A? (a) 2 B DC (b) 2CDB (c) 2 CDB (d) 2BDC (e) 2B DC McMaster University Math B3 Fall 27 Page 4 of

5 McMaster University Math B3 Fall 27 Page 5 of [ ] [ ] a b 8. Suppose that A = and B = and when multiplied together, they c d commute. Which of the following must necessarily be true? (a) a = (b) b = (c) c = (d) d = (e) none of the above 9. The following two commands are entered into Matlab: A = [ 2 3; 2 3 4; 4 5 6] A.^2 What is the output of the second command? (a) ans = (b) ans = (c) ans = (d) ans = McMaster University Math B3 Fall 27 Page 5 of

6 McMaster University Math B3 Fall 27 Page 6 of. Compute A if (A T + 3I) = [ ] 2 (a) [ ] 2 (b) [ ] 2 (c) [ ] 2 (d) [ ] 2 (e) [ 2 ].. Which one of the following statements is not equivalent to the others? (a) A is not invertible. (b) Ax = has more than one solution. (c) The reduced row echelon form of A has no row of zeros. (d) A is not a product of elementary matrices. (e) Ax = b is inconsistent for some n matrix b. McMaster University Math B3 Fall 27 Page 6 of

7 McMaster University Math B3 Fall 27 Page 7 of 2. Let be a nonzero number. What is the inverse of the matrix 2 (a) (b) (c) (d) (e) Consider the following two elementary matrices: [ ] E = E 2 = 7 [ 5 What is the matrix A that satisfies E E 2 A = I 2? (Here, I 2 is the 2 2 identity matrix.) [ ] 5 (a) [ ] 35 (b) 7 [ ] 5 (c) 7 [ ] 5 (d) 7 7 [ ] 5 (e) 7 ]. McMaster University Math B3 Fall 27 Page 7 of

8 McMaster University Math B3 Fall 27 Page 8 of 4. The system of 4 equations in 4 unnowns Ax = B has solutions 4 3 x = 9 + s + r where s and r are real numbers. If performing row operations on the augmented matrix [A B] can produce the following matrix a b c 4 3 d 5 what is a + b + c + d? (a) -27 (b) -2 (c) 2 (d) 3 (e) Not enough information provided 5. A square matrix A is called sew-symmetric if A T = A. Find all the values of the unnown constant that mae the matrix A a sew-symmetric matrix. (a) a = 2, b = 2, c = 2. A = (b) a =, b = 5 2, c = 2. (c) a =, b = 2, c = 2. 4b a + b 4b a b + c + a 4 a + c 8c b + c (d) a = 3, b = 3, c = 3. 2 (e) There is no choice of a, b and c that maes the matrix sew-symmetric. McMaster University Math B3 Fall 27 Page 8 of

9 McMaster University Math B3 Fall 27 Page 9 of 6. Which of the following statements are true? () Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation. (2) A linear system is inconsistent if it has an infinite number of solutions. (a) () is false and (2) is false. (b) () is true and (2) is false. (c) () is false and (2) is true. (d) () is true and (2) is true. 7. Which of the following statements are true? () If a matrix is in reduced row echelon form, then it is also in row echelon form. (2) There exists a 27 matrix with 27 leading ones. (a) () is false and (2) is false. (b) () is true and (2) is false. (c) () is false and (2) is true. (d) () is true and (2) is true. 8. Which of the following statements are true? () If A and B are n n matrices, then tr(a + B) = tr(a) + tr(b). (2) If A and B are n n matrices, then tr(ab) = tr(a)tr(b). (a) () is false and (2) is false. (b) () is true and (2) is false. (c) () is false and (2) is true. (d) () is true and (2) is true. McMaster University Math B3 Fall 27 Page 9 of

10 McMaster University Math B3 Fall 27 Page of 9. Which of the following statements are true? () If A and B are n n and invertible, then B A is the inverse of AB. [ ] a b (2) If A = and ad bc =, then A is not the product of elementary c d matrices. (a) () is false and (2) is false. (b) () is true and (2) is false. (c) () is false and (2) is true. (d) () is true and (2) is true. 2. Which of the following statements are true? () If A is an n n matrix and Ax = b has more than one solution for an n matrix b, then A is invertible. (2) If a diagonal matrix is invertible, then all of its diagonal entries must be nonzero. (a) () is false and (2) is false. (b) () is true and (2) is false. (c) () is false and (2) is true. (d) () is true and (2) is true. END OF TEST PAPER McMaster University Math B3 Fall 27 Page of