**Unformatted text preview: **Midterm Examination Fall 2016 Civil & Environmental Engineering
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Signature: CIVE 115: Linear Algebra Instructor: DJ Brush Date: October 20, 2016 Time: 12:30‐2:10 pm Duration: 100 minutes Pages: 9 (including cover) + separate half‐page formula sheet Notes: 1. This is an individual closed book exam. Q 2. A separate formula sheet is provided. 3. The only other aid permitted is a basic scientific calculator that meets the following restrictions: Mark 1 /18 2 /12 3 /10 4. Turn off your wireless device(s) and place it (them) in your backpack/purse. 4 /10 5. Enter your personal information above & print initials on top right of each page. 5 /15 6. Answer all questions on the exam paper using the back of pages if necessary. Clearly label solutions that span pages. 6 /15 7. Solutions must be organized and use a sufficient number of steps, so that they can be easily understood. Total /80 It cannot perform vector or matrix operations It cannot solve systems of equations It cannot store alpha‐numeric text sentences Students must fill‐in and sign the declaration above. 8. Zero or minimal marks will be awarded to answers without solutions (excluding multiple choice answers). 9. Do not separate pages. 10. Hand in exam paper and formula sheet. F16 | CIVE 115 Midterm Questions 1a) to 1f) are equally weighted (3 marks each). 1a) The unknown function f ( x) has been evaluated at three values of x in the table below. Determine if the function is linear. x 0
1
2 f ( x)
0
1/3
4/3 1b) A linear system has 5 equations with 3 variables. Is the system underdetermined, overdetermined or neither? If the equations are all independent, will the system have no solution, one solution, many solutions, or are multiple solution types possible? 1c) Explain geometrically why the trivial solution of a homogeneous linear system always exists. 2|9 F16 | CIVE 115 1d) Midterm 3|9 Explain the difference between matrix row equivalence and matrix equality, and how they are related. 1 4 1e) What are the multiplicative identities that exist for the matrix A 2 5 ? 3 6 Show how they are multiplied with A. 1f) Suppose A and B are square matrices of the same size. Indicate if each of the following statements is always true or not always true (circle one answer for each). (i) ( AB ) 2 A2 B 2 Always True Not Always True (ii) ( A B ) 2 A2 BA AB B 2 Always True Not Always True (iii) ( AB) B Always True Not Always True 1 T T AT 1 F16 | CIVE 115 Midterm 0 0 1 0 1 2
3 2 2a) Find the rank of the matrix . 1 2 2 1 1 1 2 2 1 2 2 5 2 1
2b) Given A 3 2 , B , and C , determine the following: 3 2 1
3 1 1
(i) 3B AT (ii) CT B 4|9 F16 | CIVE 115 Midterm t t t 3. Let A 1 2 3 . Find all values of t such that A1 exists. 0 1 t 5|9 F16 | CIVE 115
Midterm 1 2 13 20
4. Let B and C . Find all possible matrices A that satisfy AB C . 3 4 5 8 6|9 F16 | CIVE 115 Midterm 7|9 5. Consider the electrical network below where straight lines represent wires, the jagged lines represent resistors, and the circles with + and symbols represent batteries. The batteries supply power with fixed voltages to the circuit that cause current to flow through each resistor. Kirchoff’s current and voltage laws, and Ohm’s law can be used to determine the following three relationships between the unknown currents i1, i2 and i3 (in amps): 12 V + i1
1 ohm i1 i2 i3 0 i2 i1 2i2 12 2 ohm 1 ohm i3 2i2 i3 7 + 7 V a) Express the system of equations in the form Ax b . b) Use the inverse method to determine each current. F16 | CIVE 115 Midterm 8|9 6. The figure below shows the flow of traffic along one‐way streets during rush hour in the downtown area of a city. The arrows indicate the direction of traffic flow along each street. Measured flows are labelled as constants (in vehicles per hour) on the diagram, and unknown flows are labelled as the variables x1, x2, x3, x4, and x5. The four intersections in the area are labelled A, B, C and D. 700 200
400 x1 x5
B A x4
D
300 x2
x3 C 600
200 To avoid traffic congestion in the downtown area: all traffic entering an intersection must also leave that intersection, the minimum flow on each street must be 100 vehicles per hour, and the maximum flow on each street must be 800 vehicle per hour (maximum capacity). a) Determine the system of equations that governs uncongested traffic flow at the four intersections in the downtown area. b) Determine the general solution for x1, x2, x3, x4, and x5 that ensures no traffic congestion will occur. F16 | CIVE 115 Extra Space Midterm 9|9 ...

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