Can the Inverse of a Matrix Solve an Inconsistent System?

[amninder15]

I am having trouble solving this problem. Since in order to solve for x we need to find (A^-1)B but the problem is that in order to find A^-1 when we do the determinant I get 0 so that implies
it is inconsistent. I am not sure I have the right approach to this question.

Any help will be appreciated.

 

[LCKurtz]

The determinant being zero does not necessarily mean the system is inconsistent. The proper method to solve is to use row reduction on the augmented matrix. You may find there are infinitely many solutions.

 

[amninder15]

Ok so this is what I am doing
I used the row reduction method and I found this

[0 0 0 | -7/4]
[0 0 1 | -1/4]
[1 1 0 | 7/4]

But now the problem again is same if you look at the first row it clearly implying 0 = -7/4 which is absurd so doesn't it mean that the system is inconsistent. Or did I have done something wrong?

 

[LCKurtz]

Ok so this is what I am doing
I used the row reduction method and I found this

[0 0 0 | -7/4]
[0 0 1 | -1/4]
[1 1 0 | 7/4]

But now the problem again is same if you look at the first row it clearly implying 0 = -7/4 which is absurd so doesn't it mean that the system is inconsistent. Or did I have done something wrong?

That would indeed imply the system is inconsistent. But I think you made an arithmetic mistake somewhere in your row reduction. Hard to say where since you didn't show your steps.

 

[Mark44]

Ok so this is what I am doing
I used the row reduction method and I found this

[0 0 0 | -7/4]
[0 0 1 | -1/4]
[1 1 0 | 7/4]

But now the problem again is same if you look at the first row it clearly implying 0 = -7/4 which is absurd so doesn't it mean that the system is inconsistent. Or did I have done something wrong?

The mistake is in the top row, in the constant at the right.

 

[amninder15]

Yea I did made a mistake. Now everything looks good.
So this is my final matrix now

[1 1 0 | 7/4]
[0 0 0 | 0]
[0 0 1 | -1/4]

so that implies y = s, x = 7/4 -s and z = -1/4.


Thanks for your help really appreciate it.