MTH501 Midterm Past Paper 3

Let A = A

5 A 5 20 10

3 � 2

AB �2�6

BA 29�2

MTH501-Linear Algebra

Midterm Special 2006

Q1. 3 1

4 2

Let A= Write 5A.Is det5A=5detA?

3 1 4 2 3 115 5

det 5 A ((1510)� (20 5)) det 5 A 150� 100 50 det 5 A 50

det A ((3 2)� (1 4)) det A 6� 4 2 5det A 5 2 10

Hence proved det5A not equal to 5detA

Q2. 5 1

2 0 4 3

Let A = and B = .Show that these matrices do not commute.

Solution:-

 5 12 014 3 �3 24 3 2 05 110 2 4 33�2

AB BA

Q3. �5 4

A1 (b)

A2 (b)

x2 Use Cramer’s rule to solve the system 3x1� 2 x 2 6 �5x1 4 x 2 8

Solution:-

3x1� 2 x 2 6

�5x1 4 x 2 8

View the System as Ax =b. Using the notation introduced above.

 3�2 A

6�2 8 4  3 6 �5 8 Since detA = 2, the system has a unique solution. By Cramer’s rule,

x1 det A1 (b) det A 24 16  20

det A2 (b) det A 24 30  27

Q4. Find the matrix of linear transformation T : R3� R4 defined by

T ( x1 , x2 , x3 ) x1 x2 , x2 x3 , x1� x3 , x1 3 with respect to the standard basis of R ?

Solution:-

The standard basis of R3 is {e1, e2, e3} E1 = (1, 0, 0) E2 = (0, 1, 0) E3 = (0, 0, 1)

T(e1) = T (1, 0, 0) = (1, 0, 1, 1) T(e2) = T (0, 1, 0) = (1, 1, 0, 0) T(e3) = T (0, 0, 1) = (0, 1, -1, 0) The matrix of linear transformation is 16

6�3

R� R be a linear transformation defined by T(x)=Ax , where

3 5 2 11

8 38

47 45

c) d) None of these

Determine whether the set S={ �42

} is a spanning set for R or not.

Let T : 1� 3

�1 7 3

�5

A= ,then find an x in R whose image under T is b.Where b=

1 2 5 1 The inverse of the matrix  1� 2 �5 1 (a) 1 1� 2 9 5 1 b) is

1�1 9 5 2 �1

(c)

d) None of these

The set of vectors {(5,0,0), (7,2,-6), (9,4,-8)} is, a) Linearly independent

b) Linearly dependent

c) Basis of R3