# Thread: Determining inconsistent system

1. ## Determining inconsistent system

Is there anyway to determine at face-value whether or not this linear system of equations is consistent or not? Consistent being that it has at least one solution. For example...

I would really appreciate help!!!

- Olivia

2. ## Re: Determining inconsistent system

If a square matrix of coefficients is non-singular there will be a unique solution.

A non-singular matrix will have a non-zero determinant.

This matrix is fairly sparse so computing the determinant via minors shouldn't be that difficult.

Of course you can just dump it into Wolfram as well. It should be

Det[{{1,0,3,0},{0,1,0,-3},{0,-2,3,2},{3,0,0,7}}]

on the command line

3. ## Re: Determining inconsistent system

I don't know what you mean by "determine at face value". If you mean "just look at it an immediately know", I can't help you. I am just not that good. I would have to actually do the calculations!

The system of equation is
$x_1+ 0x_2+ 3x_3+ 0x_4= 2$
$0x_1+ x_2+ 0x_3- 3x_4= 2$
$0x_1- 2x_2+ 3x_3+ 2x_4= 1$
$3x_1+ 0x_2+ 0x_3+ 3x_4= -5$.

As romek said, that is fairly sparce- there are a whole lot of 0s in there.

We can get even more 0s by:
1) Subtract 3 times the first equation from the fourth and add twice the second equation to the third:
$x_1+ 0x_2+ 3x_3+ 0x_4= 2$
$0x_1+ x_2+ 0x_3- 3x_4= 3$
$0x_1+ 0x_2+ x_3- 4x_4= 7$
$0x_1+ 0x_2- 9x_3+ 7x_4= -11$.

That is almost triangular.
2) Just add three times the third of those equations to the fourth to get
$x_1+ 0x_2+ 3x_3+ 0x_4= 2$
$0x_1+ x_2+ 0x_3- 3x_4= 3$
$0x_1+ 0x_2+ x_3- 4x_4= 7$
$0x_1+ 0x_2+ 0x_3- 5x_4= 10$.

Which is triangular and it is easy to see that is a consistent set of equations. In fact, it is easy to solve the system by "back-substituting".

4. ## Re: Determining inconsistent system

ok ok thank you