What are the conditions for a unique solution in a matrix?

[Lily@pie]

Hi!

Homework Statement

Given augmented matrix
x y z | 0
0 l k | 1
0 w b | 9

x is non zero

Find the conditions that involve x,y,z,l,k,w,b so system is 100% has an unique solution.

2. The attempt at a solution
I know that for unique solution, there must be no free variable and no row [ 0 0 0|a ], a=nonzero
I've tried a few ways.

(I) As there must be no free variables, each row must have a leading entry. Hence I reduce it to
x y z | 0
0 1 k/l | 3/l
0 0 b-(wk/l) | 4-(3w/p)

According to this, b-(wk/l) /= 0
However, this condition doesn't include x,y and z.

(II) I also derived a condition to produce an identity matrix for the coefficient matrix.
x=l=b=1
y=z=k=w=0
This is the only condition i get that involves all the unknown.

(III) The last idea I have in mind is to reduce the whole matrix to it's reduced row echelon form. However, the unknowns will be all in the constant matrix when the coefficient matrix is reduced to the identity matrix. It's weird ><

Thanks SO SO much

 

[vela]

Are you familiar with determinants?

 

[Lily@pie]

Erm... I learned it in high school before... Not too deap

 

[vela]

OK, I just brought it up because if you understood the significance of the determinant, this problem is pretty easy.

Your first method is a good start. Note that by dividing by l, you're assuming l≠0. You still need to consider the case when it is equal to 0. Also, you can multiply what you have so far by l to get lb-wk≠0. It just looks a little nicer.

The fact that there are no restrictions on x, y, and z (other than you were given x≠0) isn't really a problem. It just means that regardless of what they are equal to, you can find a unique solution.

 

[Lily@pie]

Oh, that means I started off correctly but my 2nd and 3rd idea is wrong? It is not necessary to include x,y and z in the condition if it has no restriction. Right?

When I assume l = 0, isn't it infinite solution? Because there will be free variables. Just to make sure my concept is correct.

 

[vela]

The second idea is definitely wrong, and the third approach is essentially the same as the first one.

It's possible to have l=0 and a unique solution. For example, the system

\left(<br /> \begin{array}{ccc|c} 1 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 1 &amp; 1 \\ 0 &amp; 1 &amp; 0 &amp; 9<br /> \end{array}<br /> \right)

clearly has a unique solution.

 

[Lily@pie]

OOHHH! True too!

But the question specifies that to derive a condition that must be in form of equation that involve all the unknowns doesn't matter?

I am so sorry because english is not my first language.

Thanks so much for your help too! You are my savior!

 

[vela]

No, I think the problem is asking you for conditions that involve any of those variables, but not necessarily all of them.