DAWSON COLLEGEDEPARTMENT 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. Scientic 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.

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2(1) [4 marks]Solve using Gauss-Jordan elimination.

x1 x2 + x3 + 2x4 = 12x1 x2 + 3x4 = 3

3x1 2x2 + x3 + 5x4 = 4x1 + x3 x4 = 2

3(2) [4 marks]Find all values of k so that the homogeneous system

(5 k)x+ y = 02x+ (4 k)y = 0

has innitely many solutions.

(3) [4 marks]Let A1 =

0B@ 1 2 00 1 21 0 1

1CA and B =0B@ 0 1 11 0 1

0 1 0

1CA. Simplify rst andthen compute B1(2ATB2)1.

4(4) [4 marks]Given A =

0B@ 1 2 00 1 11 0 3

1CA and XA = 2 0 11 0 2

!, nd X.

(5) [4 marks]Denition: A matrix A is called self-inverse if A1 = A. Let

A =

3 x2 3

!. Find x so that A is self-inverse.

5(6) [4 marks]If det(A) = 3 and A is 4 4, nd det(2A1 + 5adj(A)).

(7) [4 marks] If A and B are 4 4 matrices with det(A) = 2 and

AB =

0BBB@1 0 2 12 1 0 4

1 0 1 21 1 0 0

1CCCA, nd det((2B)1AT ).

6(8) [4 marks]If

a b c

d e f

g h i

= 2, nd2a 3b 2d 3e 2g 3h

b e h

c b f e i h

(9) [4 marks]Use Cramer's rule to solve the system

2x 3y = 15x+ 4y = 7

7(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 R3 such that !u +!v +!w = !0 , then

!u !v = !v !w = !w !u

8(12) [4 marks]If !u and !v are non-zero vectors, prove that (!u +!v ) perpendicularto (!u !v ) if and only if k!u k = k!v k.

9(13) [4+4+4 marks]Given the two lines L1 :

8>:x = 2ty = 1 + 2tz = 3t

and L2 :

8>:x = 6 + 4s

y = 3 + s

z = 1 s(a) Find the point of intersection of L1 and L2.

(b) Find, in general form, an equation of the plane containing L1 and L2.

(c) Find parametric equations of the line perpendicular to both L1 and L2and passing through the point B(2; 1; 2).

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(14) [4+4+4 marks]Given the line L :

8>:x = 2 + 3t

y = 3 + t

z = 1 tand 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?

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(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]LetM22 be the vector space of all 22 matrices with the standardoperations of addition and scalar multiplication, andW = fA 2M22 j tr(A) = 0g.Note: tr(A) =sum of elements in the main diagonal of A.

(a) Show W is a subspace of M22.

(b) Find a basis for W and state the dimension of W .

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(17) [4+4 marks]Let !u = (1; 2; 1; 1); !v = (0; 1; 2; 1) and !w = (2; 3; 0; 1).(a) Is (2;1;8;3) in spanf!u ;!v ;!w g?(b) Are !u ;!v ;!w linearly independent?

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(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 A3 3A = 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 .