MATH 218 Exam 1, Fall 2017

Notes before solving the exam: A-PDF Watermark DEMO: Purchase from www.A-PDF.com to remove th 1) You have to solve the recommended problems in the book after understand- ing each chapter from the book and the notes. 2) If you have any questions or con- October 9, 2017 MATHEMATICS 218 NAME cerns, let us know through our mail: in- Time : 55 minutes QUIZ I ID# — sightclub@gmail.com. Fall 2017-18 GOOD LUCK :) Circle your section number : Hazar Abu-Khuzam | Sabine El Khoury Sabine El Khoury Michella Bou Eid 1 |? 3 4 5 6 7 8 9 10 ey) 12 2w [iw iw /2M [iM |[1iM [4M [3M | 2Th nF [@F) TE 1 PROBLEM GRADE PARTI } Y 1 126 PART II 5|6|7/8|9 |10| 11 aja|ajaja a & vb [vo 6 |b Ly |® Tb £ iS) c FO |e Kay] d djd [dq |@& ejejelsele e e 5-11 9S- / 28 PART II 9) eats 14 Van 16 7 7 ¥ ©xr @® lr ig F ome 9) /\ 12-18 = / 21 | TOTAL LW 7100 PARTI. Answer each of the following problems in the space provided for each problem ( Problem 1 to Problem 4). 1. Find the values of a and b for which the following system xty+z=b _ - 3x -3y +-az=8 _— 7) xt+y+z=1 j ale insight has e tub a. no solution b. a unique solution c. infinitely many solutions. [ 14 points] 2. Consider the matrix 001 —s of | insight (a) Find (A YQ"! [ 5 points] -\ rN ye (a\ b ¢ 3. Show that for all real numbers a, b and c , the vector | fi) fA) vectors | 1 |, 2 and 8 (0 0 5 can be written as a linear combination of the 4. Let A and B be 2x2 matrices such that AB+2A = -I. (a) Find an expression of A” in terms of B A(B+ar)=- [ 8 points] a 0 (b) Suppose that the above matrices A and B are given to be A =B= for some a nonzero number a. Find all possible values of a. [7 points] PART IL. Circle the correct answer for each of the following multiple choice problems ( Problem 5 to Problem 11) IN THE TABLE IN THE FRONT PAGE. [4 points for each correct answer]. 5. Let A be a 3x3 matrix such that A* = A. Then, a. A is invertible b. det (A) =0 ce A=I q @ 4=2 PKA = AAAZK-A e, None of the above. UN \ [ 4 points] 6. y), g If A and Bare invertible 3 x3 matrices suchthat O\ « det(2A) = 2 = det(4?(B")') 2 , x % Then, ig a. det (B)=1/2 g sex) ~ a v y) oe» b. det (B)=2 pr Va © det (B)= 32 7 \ q . det (B)=8 ) VA e. None of the above. 7, Which one of the following statements is TRUE? a. If AB = AC, then B= C. b. If AB= 0 then A = 0 or B= 0. c. If A is invertible then Ax = x has only the trivial solution for x. @ifb can be written as a linear combination of the columns of A then AX = b is consistent. (A-) x = O [ 4 points] 8. Let A be an invertible nxn matrix. Which one of the following statements is FALSE: AS Atis invertible. ) The number of nonzero rows in a row echelon form of A is n. AB is invertible for any nxn matrix B. det (A) #0. Sy The reduced row echelon form of A is I. [ 4 points] 9. Let vi, v2, V3, v4, and vs be vectors in R" such that v3= vit v2, and vs= vstvs.. Which one of the following statements is FALSE: a7) Ys is a linear combination of v1, v2, and v4 v2 is a linear combination of v1 and v3 vi is a linear;combination of v4and vs @ vs isa linear combination of va and vs 6) vs is a linear combination of vi and v2 [ 4 points] 10. Let A be an nxn matrix. Which one of the following statements is FALSE: If the reduced row echelon form of A is I, then A is invertible @y if A is not invertible, then the matrix equation AX=b has infinitely many solutions for all b. oe the homogeneous matrix equation AX=0 has only the trivial solution, then A is invertible Ifthe reduced row echelon form of A is not I, then det(A)=0 {e) If A is invertible, then the homogeneous matrix equation AX=0 has only the trivial solution [ 4 points] be a c b if A=|d e f | suchthat |A|=5,then Bat+d 3c+f 3b +e} is equal to: ghi 2g 2i 2h (@) -30 i % b (b) 30 (c) 10 a e& + C 10 ’ nN e) None of the above i) Cc [ o\ = 4 d cs Y * PART IIL Answer TRUE or FALSE only, IN THE TABLE IN THE FRONT PAGE (3 points for cach correct answer) &)sA Pr 12. If A is any square nxn matrix, then A'A is symmetric. 4 (a> ie pe 13. Every triangular matrix is invertible. ¥ 14. IfA and Bare nxn matrices such that AB=0 such that AZ0, then B=0 . € 15. IfA and B are nxn matrices such that A is invertible and (A“'B) is invertible, then B is invertible. \ 3 w) 1 17. IfA is a 3x3 matrix such that det 2A“) =4, then det(A)=2. 2 ] 0 T 16. The vector | 1 | is a linear combination of the vectors| 0 | and | —1 18. If Bis a2x2 matrix such that B’ = —B, then B is not invertible x [21 points]