Linear Systems Question

Question

Let t ∈ R and consider the linear system described by the following augment matrix:

A = [ 1 -1 | 0 ]

[ 1 1 | 1 ]

[ 0 1 | t ]

For which values t does this system have:

(i) no solutions

(ii) a unique solution

(iii) infinitely many solutions

Answer 1

First, you want to row reduce. You can also turn the problem into a system of equations without row reducing.

A = [1 -1 | 0] = [-1 1 | 0 ]

[1 1 | t] [0 2 | 1 ]

[0 1 | t] [0 0 | t - 1/2]

Now, if the first column is x, and the second column is y:

-x + y = 0; y = x

2y = 1; y = 1/2

0x + 0y = t - 1/2

IF t = 1/2, we have a unique solution (1/2, 1/2)

IF t ≠ 1/2, we have 0 ≠ 0, so there are no solutions.

Hence, these are all the possible values of t. No value of t results in infinitely many solutions.