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DAWSON COLLEGE
MATHEMATICS DEPARTMENT
Final Examination
Mathematics 201-NYC-05 Date: Tuesday, May 18, 2010 Linear Algebra (Regular) Time: 2:00 - 5: Instructors: M´elanie Beck, Yann Lamontagne
- Consider the following system of equations:
9 x − 18 y + 45 z = − 39 − 2 x + 5 y − 11 z = 28 / 3 7 x − 17 y + 38 z = − 97 / 3
(a) (5 marks) Find all solutions using Gauss-Jordan elimination. (b) (2 marks) Find any two particulars solutions.
- Consider the matrices
A =
[
]
B =
[
]
C =
[
]
D =
[
]
Compute whenever it is possible: (a) (2 marks) A−^1 (b) (2 marks) C−^1
(c) (2 marks) (3A − I)B − CT (d) (2 marks) det(3B)
(e) (2 marks) trace(AC)T
- Let A =
and B a 4 × 4 matrix with det(B) = 3.
(a) (3 marks) Find det(A). (b) (3 marks) Find det(− 3 A−^1 B^2 ).
- (5 marks) Prove that if AT^ A = A, then A is symmetric and A = A^2.
- (5 marks) Show that if a square matrix A satisfies A^2 − 4 A + I = 0, then A is invertible and A−^1 = 4I − A.
- (4 marks) Solve for x:
x 0 3 0 x + 1 5 0 0 2
−x 5
- Consider the matrices
A =
[
]
B =
[
]
C =
[
]
(a) (2 marks) Is it possible to find an elementary matrix E 1 such that E 1 A = B? If yes, what is E 1? If no, justify. (b) (2 marks) Is it possible to find an elementary matrix E 2 such that E 2 B = C? If yes, what is E 2? If no, justify.
- Consider the following system of equations:
3 x + 2 y = − 12 −x + 4 y = 7 ,
(a) (3 marks) Solve the system using Cramer’s rule. (b) (3 marks) Solve the system using the inverse of A.
- Given u = (3, 0 , 1), v = (− 2 , 1 , 2) and w = (4, − 2 , 1), find
(a) (2 marks) ‖u × v‖ (b) (2 marks) the cosine of the angle determined by u and v (c) (2 marks) projv (2w) (d) (2 marks) a unit vector perpendicular to u + v and to w.
- Let u = (1, 4) and v = (k, 2). Find k such that
(a) (3 marks) u and v are parallel (b) (3 marks) u and v are perpendicular
- (5 marks) Find an equation for the plane that passes through the point (3, − 6 , 7) and is perpendicular to the planes 5x − 2 y + z − 5 = 0 and 3x + 4y − z + 6 = 0.
- (5 marks) Find an equation for the plane, each of whose point is equidistant from (− 1 , − 4 , −2) and (0, − 2 , 2).
- (5 marks) Find the area of the triangle with vertices A(− 3 , 1), B(2, 4), C(1, −2).
- (5 marks) Using projection(s), find the distance between the point (0, −2) and the line 3x − 2 y + 5 = 0 (that passes through (− 1 , 1) and (− 3 , −2)).
- (5 marks) Find an equation of the line that passes through the origin and is perpendicular to the lines
L 1 : x + 3 = +4t, y = −t, z − 1 = 2t, −∞ < t < ∞,
L 2 : x = 3 + 5t, y = − 2 , z = − 6 − 7 t, −∞ < t < ∞.
- (5 marks) Using projection(s), find the distance between the following planes:
Plane 1: 5x + 2y − 3 z = 7, Plane 2: − 10 x − 4 y + 6z + 10 = 0.
- (9 marks) Solve the following linear programming problem:
Maximize P = 5x + 4y + 3z subject to 2 x + 3y + z ≤ 5 4 x + y + 2z ≤ 11 3 x + 4y + 2z ≤ 8 x ≥ 0 , y ≥ 0 , z ≥ 0
Answers:
- a. x = − 3 − 3 t, y = 2/3 + t, z = t. b. Replace t in a. by any value to get a particular solution.
2. A−^1 =
[
]
b. Not possible c.
[
]
d. Not possible e. Not possible.
- a. −3 b. − 243
- x = 4 or x = − 3
- a. Yes and E 1 =
[
]
b. No
- x = − 31 /7 and y = 9/ 14
- a.
74 b. −^2
√ 10 15 c. (^
32 9 ,^
− 16 9 ,^
− 32 9 ) d.^ √^1 206
- a. k = 1/2 b. k = − 8
- − 2 x + 8y + 26z − 128 = 0
- x + 2y + 4z + 13/2 = 0
- 27/
- 9/
- x = 7t, y = 38t, z = 5t
- 2/
- x = 2, y = 0, z = 1 (P = 13).
