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MATH204 Final EXAM WITH Solutions DEC 2018
Vectors and Matrices (Concordia University)
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CONCORDIA UNIVERSITY
Department of Mathematics & Statistics
Course Number Section(s)
Mathematics 204 All except EC
Examination Date Pages
Final December 2018 2
Instructors Course Examiner
All E. Cohen
Special Instructions:
[> Only approved calculators are allowed.
l> Justify all your answers.
[> All questions have equal value.
MARKS
1. Use Cramer’s rule to compute the solution of the system:
II’] + 1‘2 2 3
—3ml + 2.13 = 0
:1'2 — 2.173 = 2
l 0 —2
2. Find the inverse of the matrix A 2 ~13 1 4 , if it exists.
2 —3 4
3. Find all solutions of the system:
1:1 + 61'2 + 2273
213
~ 51‘;
— 81:4
—
—
2:175
:115
:
2
—-4
I5 :— 7
1 3 0 5
2 7 1 3
4. Flnd the detemnnant of.
_
.
. A —
1 2 1 6
2 3 4 5
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MATH 204 Final Examination December 2018 Page 2 of 2
5. a) Let u : (1, 5, —2). v = (3. —1. 5). Find the orthogonal projection of v on u.
b) Let 2 (3.1.2),
u': (2,3,1), 1L2 113 2 (1.2.3). Find Che-2c; such that
01 U1 +02 11.2 +C3 U3 = (1,0,1).
6. a) Find the area of a triangle with vertices (1, 1, 2) , (0.1.4), (1. 2, 5).
Find a vector orthogonal to the plane of the triangle.
b) Find the distance between the point (2, —3) and the line 217 2 3y + <1.
7. a) Are the vectors (2. ——2, 1) . (1, —3. 2) . (—7.5,4)
linearly dependent or independent?
b) Find the parametric equations for the line in R3 passing through
(1. 4, 5) and perpendicular to the plane 23* — 41/ + 3; = 1.
@HI
7
8. Let A: OOH COCO
GOA Ol-‘O OCEN)
H00
5 and X =
NM
—2 u
v
117
Find a basis for the solution space of the homogeneous system AX = 0.
9. Find the standard matrices for following operators on R2:
a) a rotation clockwise of 45°.
b) a reflection about line y = ~17.
l 3 3
10. Let A = —3 —5 —3 . Find a matrix P such that P”1AP = I).
3 3 1
a diagonal matrix.
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of the present document may be sold. reproduced. republished or i'e»Llisseiniiiate<l in
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holder. copyright
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