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Linear Algebra Exam: Gauss-Jordan, System Solutions, Matrix Mult., and Vector Projection, Exams of Linear Algebra

English and Foreign Languages UniversityLinear Algebra

The solutions to a linear algebra final exam from dawson college, mathematics department. The exam covers topics such as solving linear systems using gauss-jordan method, determining the number of solutions, matrix multiplication, and vector projection.

Typology: Exams

2012/2013

Uploaded on 02/14/2013

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1
Dawson College, Mathematics Department
Final Examination
Linear Algebra 201-NYC-05 (Commerce)
(Sections 6, 7)
December 17, 2009
T. Kengatharam, V. Ohanyan
1. (6 marks) Solve the linear system by Gauss-Jordan method.
522
19752
1243
4321
4321
4321
xxxx
xxxx
xxxx
Ans.
txxtxtx 4321 ,3,2,31
2. (6 marks) For which values of k the system has
a) Exactly one solution, b) No solution, c) Infinitely many solutions
834
8352
32
kzkyx
zyx
zyx
Ans. a)
0k
b)
0k
c) never
3. (5 marks) Let
A
be an invertible matrix, such that
AIAA 232 23
. Show that
IAAA 232 21
Ans.
IAAAIAAAAA
232)232( 2321
IAAAAIAAAA
232)232( 2321
pf3

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Dawson College, Mathematics Department

Final Examination

Linear Algebra 201-NYC-05 (Commerce)

(Sections 6, 7)

December 17, 2009

T. Kengatharam, V. Ohanyan

  1. (6 marks) Solve the linear system by Gauss-Jordan method.

1 2 3 4

1 2 3 4

1 2 3 4

x x x x

x x x x

x x x x

Ans. x 1 (^)  1  3 t , x 2  2  t , x 3  3 , x 4  t

  1. (6 marks) For which values of k the system has

a) Exactly one solution, b) No solution, c) Infinitely many solutions

x y k z k

x y z

x y z

Ans. a) k  0 b) k  0 c) never

  1. (5 marks) Let A be an invertible matrix, such that 2 A 3 A I 2 A

3 2   . Show that

A 2 A 3 A 2 I

1 2   

Ans. AAA AAIAAAI

 ( 2 3 2 ) 2 3 2

1 2 3 2

A A  A  A  I A  A  A  A  I

 ( 2 3 2 ) 2 3 2

1 2 3 2

  1. The following linear system is given.

 

x y z

x y z

x y z

a) (5 marks) Find the inverse of the coefficient matrix

b) (3 marks) Use the result found in part a) to solve the system

Ans. a)

b) x  1 , y  3 , z  6

  1. (5 marks) Use the Cramer’s rule to solve the following system for “y” only.

x y z

x y z

x y z

Ans. y  3

  1. (5 marks) Find the matrix X , if

X

Ans. 

7. Let AandB be ^2  2 matrix, such that det ^ A ^  2 ,det^ B ^  3. Find

a) (4 marks)  

T A B

1 det

b) (4 marks) ^ ^  

1 det 2

A B

c) (4 marks)det   B adj  3 A 

d) (5 marks) Let  5

g h i

d e f

a b c

. Find

f d i g c a

e d h g b a

d g a

Ans. a) 2

b) 3

c)  54 d) 30

  1. (3 marks) Let u andv

be two vectors, such that u  v  1 , 2 , 3 

. Find the vector

 u v   v u 

  2  Ans.  3 , 6 , 9 