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DAWSON COLLEGE
DEPARTMENT OF MATHEMATICS
FINAL EXAMINATION
LINEAR ALGEBRA 201-NYC-05 (Science)
Fall 2010 Time: 3 hours.
Instructors:D. Dubrovsky; I. Gombos; C. Gowrisankaran; T. Kengath-
aram; V. Ohanyan; B. Szczepara
Name:
ID:
Instructions:
- Translation and regular dictionaries are permitted.
- Scientific non-programmable calculators are permitted.
- Print your name and ID in the provided space.
- This examination booklet must be returned intact.
This examination consists of 18 questions. Please ensure that you have
a complete examination before starting.
1
(1) [4 marks]Solve using Gauss-Jordan elimination.
- x 1 − x 2 + x 3 + 2x 4 =
- 3 x 1 − 2 x 2 + x 3 + 5x 4 = - −x 1 + x 3 − x 4 = −
(4) [4 marks]Given A =
and^ XA^ =
, find X.
(5) [4 marks]Definition: A matrix A is called self-inverse if A
− 1 = A. Let
A =
− 3 x
. Find x so that A is self-inverse.
(6) [4 marks]If det(A) = 3 and A is 4 × 4, find det(2A
− 1
(7) [4 marks] If A and B are 4 × 4 matrices with det(A) = 2 and
AB =
, find det((2B)
− 1 A
T ).
(10) [4 marks]Find the volume of the parallelepiped determined by the vectors
u = (1, 1 , 1),
v = (− 1 , 2 , 0) and
w = (0, 2 , 3) (or equivalently
u =
i +
−→ j +
k ,
v = −
i + 2
j and
w = 2
j + 3
k )
(11) [4 marks]Prove: If
u ,
v and
w are any three vectors in R
3 such that
u +
v +
w =
0 , then
u ×
v =
v ×
w =
w ×
u
(12) [4 marks]If
u and
v are non-zero vectors, prove that (
u +
v ) perpendicular
to (
u −
v ) if and only if ∥
u ∥ = ∥
v ∥.
(14) [4+4+4 marks]Given the line L :
x = 2 + 3t
y = 3 + t
z = 1 − t
and the point B(− 3 , 0 , 5).
(a) Find the point on line L closest to B.
(b) Find the distance from point B to line L.
(c) At what point does the line L intersect the xy-plane?
(15) [4 marks]Find parametric equations for the line of intersection of the planes
x + 2y + z = 0 and 2x + 3y − z = 4.
(16) [4+4 marks]Let M 22 be the vector space of all 2×2 matrices with the standard
operations of addition and scalar multiplication, and W = {A ∈ M 22 | tr(A) = 0}.
Note: tr(A) =sum of elements in the main diagonal of A.
(a) Show W is a subspace of M 22.
(b) Find a basis for W and state the dimension of W.
(18) [4+4 marks]TRUE OR FALSE? Justify your answer by giving a proof if the
statement is true or a counterexample if the statement is false.
(a) If A is an n × n matrix such that A
3 − 3 A = In, then A is invertible.
(b) If V is a vector space of dimension n and
v 1 ,
v 2 , · · ·,
v (^) n are n non-zero
vectors in V , then
v 1 ,
v 2 , · · ·,
v (^) n form a basis for V.
