Question

Chaney, Jessica D (2449) Pay Cashier Math 230 Linear Algebra Quiz- Due Monday February 1. Consider the linear system whose augmentęd matrix is of the form 14 312 1 3 alb a) For what values of a and b will the system have infinitely many b) For what values of a and b will the system be inconsistent

Answer 1

Let A =

1	1	3	2
1	2	4	3
1	3	a	b

To answer the questions, we will first reduce A to its RREf as under:

Add -1 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row

Add -2 times the 2nd row to the 3rd row

Add -1 times the 2nd row to the 1st row

Then the RREf of A is

1	0	2	1
0	1	1	1
0	0	a-5	b-4

a). Now, if a = 5 and b = 4, the solutions to the given system will be x+2z = 1 or x = 1-2z and y+z = 1 or, y = 1-z so that (x,y,z)^T = (1-2z,1-z,z)^T= (1,1,0)^T +z(-2,-1,1)^T. Thus, the given system will have infinite solutions if a = 5 and b = 4.

b). If a = 5 and b ≠4, then the last row of the augmented matrix will be (0,0,0,1) which means 0 = 1. Then the given system will be inconsistent. Thus, the given system will be inconsistent if a = 5 and b ≠4.