MATH 220 MIDTERM EXAMINATION OCTOBER 17, 2001Name ID # Section #There are ??multiple choice questions. Each problem is worth 5 points. Four possible answersare given for each problem, only one of which is correct. When you solve a pro blem, notethe letter next to the answer that you wish to give and blacken the corresponding space onthe answer sheet. Mark only one choice; darken the circle completely (you should not beable to see the letter after you have darkened the circle).THE USE OF CALCULATORS DURING THE EXAMINATION IS FORBIDDEN.CHECK THE EXAMINATION BOOKLET BEFOREYOU START. THERE SHOULD BE ?? PROBLEMSON ?? PAGES (INCLUDING THIS ONE).MATH 220 MIDTERM EXAMINATION PAGE 21. If1 2 −35 h 4is the augmented matrix of a system of linear equations, then for what value of h is thesystem inconsistent?a) h = 0.b) h = 5.c) h = 10.d) h = 15.2. Which of the following matr ices is in row echelon (but not necessarily reduced row echelon)form?A =1 0 00 1 00 2 0, B =0 0 10 0 00 0 0, C =0 1 01 0 00 0 0.a) A and C, but not B.b) A and B, but not C.c) B only.d) B and C, but not A.MATH 220 MIDTERM EXAMINATION PAGE 33. If A =1 1 2 8−1 −2 3 13 −7 4 10, then find the matrix in reduced row echelon form that is rowequiva lent to A.a)1 0 0 00 1 0 10 0 1 2b)1 0 0 30 1 0 10 0 1 2c)1 0 0 10 1 0 20 0 1 3d)1 0 0 50 1 0 70 0 0 64. If v1=11−2and v2=21−1, then for what value of h does the vector v3=75hlie inthe plane generated by v1and v2?a) h = 4.b) h = −4.c) h = 8.d) h = −8.MATH 220 MIDTERM EXAMINATION PAGE 45. What is the solution set of the following system of linear equations?−2x2+ 3x3= 12x1+ 4x2− x3= 2−6x1− 12x2+ 3x3= 5a) x1= 1 , x2= 0 , x3= −1.b) x1= 3 − 2t, x2= 2 + t, x3= tc) x1= 2 + 2t, x2= t, x3= 3.d) There are no solutions. The system is inconsistent.6. If A =1 5 −2 32 −3 1 40 2 4 6and v =2−16−3, then what is the second entry of Av?a) 1.b) 2.c) 3.d) 4.MATH 220 MIDTERM EXAMINATION PAGE 57. If A =1 −6 −32 12 2−1 −3 0, then what geometric figure is formed f r om the span of the columnsof A?a) A line.b) A plane.c) All of R3.d) All of R2.8. If A is a 4 by 4 matrix whose columns span R4, then which of the following statements isfalse?a) The equation Ax = b has at least one solution for every b.b) The columns of A are linearly independent.c) The equation Ax = 0 has a nontrivial solution.d) The linear transformation x 7→ Ax is onto.MATH 220 MIDTERM EXAMINATION PAGE 69. If A =4 −8−3 62 −4is the coefficient matrix of a system of linear equations, then what is thesolution set for the associated homogeneous linear system?a) x1= 1 , x2= 2.b) x1= 0 , x2= 0.c) x1= 2 x2, x2= t.d) There are no solutions because the the system is inconsistent .10. If T is the linear transformation defined by the formulaT (x1, x2) = (x2, −x1, x1+ 3x2, x1− x2),then what is the standard matrix for T ?a)0 −1 1 11 0 3 −1b)1 0 3 −10 −1 1 1c)0 1−1 01 31 −1d)−1 00 13 −1−1 1MATH 220 MIDTERM EXAMINATION PAGE 711. If T : R2→ R2is the linear transformation that first rotat es points clockwise by π/4 aboutthe origin and then reflects points about the x1-axis, then what is t he standard matrix forT ?a)1√21 −1−1 −1.b)1√2−1 11 1.c)1√21 11 −1.d)1√21 11 1.12. IfA =5 21 43 0and B =−1 23 −1,then what is AB?a)1 811 −2−3 6.b)1 82 2−3 6.c)−1 −85 016 3.d)−1 −83 43 −6.MATH 220 MIDTERM EXAMINATION PAGE 813. If A =2 30 4and If B =2 10 h, then for what value of h does AB = BA?a) h = 1.b) h = 2.c) h = 8/3.d) h = 3/8.14. If A =8 6−9 −7, then what is the inverse of A?a)−7 −69 8.b) −12−7 −69 8.c)12−7 −69 8.d)7 69 8.MATH 220 MIDTERM EXAMINATION PAGE 915. If A =2 0 00 −1 01 1 1, then what is the inverse of A?a)1/2 0 00 −1 0−1/2 1 1.b)1/2 0 00 −1 00 1 1.c)1/2 0 00 1 01/2 1 1.d)−1/2 0 00 1 01/2 1 1.16. If A =−1 12 1, then what is A3?a)−1 18 −1.b)7 510 −7.c)−3 36 3.d)1 −1−8 1.MATH 220 MIDTERM EXAMINATION PAGE 1017. Which of the following statements is true?a) If A is an invertible matrix, then the columns of A are linearly independent.b) If A is an n by n matrix that has n pivot positions, then the equation Ax = 0 has anontrivial solution.c) If ATis not invertible, A is not invertible.d) If A is an n by n matrix, then the linear transformation x 7→ Ax is one-to-one, thenA is invertible.18. If (AT)−1=−3 −15 2, then what is A?a)2/5 −11/5 −3/5.b)−2 5−1 3.c)2/5 1/5−1 −3/5.d)−3 5−1 2.MATH 220 MIDTERM EXAMINATION PAGE 1119. Which of the following is a linear subspace of R3?a) {(x1, x2, x3) : x1+ x2+ x3= 0}.b) {(x1, x2, x3) : x1+ x2+ x3≥ 0}.c) {(x1, x2, x3) : x1+ x2+ x3= 1}.d) {(x1, x2, x3) : x1+ x2+ x3= x1x2x3}.20. Which of the following is not a linear subspace of R3?a) The null space of a 3 × 4 matrix.b) The column space of a 3 × 4 matrix.c) The null space of a 4 × 3 matrix.d) The set of solutions of a homogeneous system of 4 equations in 3