# Homework Help: Linear system with parameter

1. May 28, 2013

### Felafel

1. The problem statement, all variables and given/known data

I have done this exercise, but I don't have a file with the solutions. COuld you please check it?

Given the following system:
$\lambda \in \mathbb{R}$

$x − z = \lambda$
$x + y + 2z + t = 0$
$y + 3z =$
$x + z + t = 0$

1-find $rk(A_{\lambda}) and rk(A_{\lambda}, B_{\lambda})$ according to the different values of $\lambda$

2- for which values of $\lambda$ does tha system have solutions?

3- what is its solutions set?

4- for which values is the system homogeneous ?

3. The attempt at a solution

This is the matrix $(A_{\lambda}, B_{\lambda})$
( 1 0 1 -1 λ)
( λ 1 2 1 0)
( 0 1 3 0 λ)
( 1 0 1 λ 0)

doing some row reduction i get:

( 1 0 -1 0 λ )
( 0 0 2 λ -λ )
(λ-1 0 0 1 -2λ)
( 0 1 3 0 λ)

and i see rk(A)=rk(A,B) for any $\lambda$, so according to Rouchè-Capelli's theorem the system has solutions.

So, answers to 1 and 2 are: 1- rk(A)=rk(A,B) for any $\lambda \in \mathbb{R}$ 2-$\forany \lambda \in \mathbb{R}$

Now, if $\lambda$=1 i get:
(1 0 -1 0 1)
(0 0 2 1 -1)
(0 0 0 1 -2)
(0 1 3 0 1) thus: x=3/2, y= -1/2, z=1/2, t=-2 is the only solution

Doing the same, if $\lambda$=0 i get (x, y, z, t)=(0, 0, 0, 0)
(IS IT ACCEPTABLE AS A SOLUTION?)

If $\lambda$ is different from 0 and 1,

x-z=$\lambda$
2z+λt=-λ
(λ-1)x+t=-2λ
3z+y=λ

with z= $\alpha$
$\Sigma$: (x,y,z,t)= (λ+$\alpha$, λ-3 $\alpha, \alpha, \frac{-λ-2\alpha}{λ}$)

so answer to nu,ber 3: the solutions sets are those above
4- the system is homogeneous for λ=0. for its solutions set, see above.

2. May 29, 2013

### Felafel

I know it is a very long exercise, so I'll just ask again for the most urging question: could anyone please tell me if $\lambda=0$ is right?
thank you :)

3. May 29, 2013

### Staff: Mentor

What's on the right side of the third equation? λ?
Your matrix doesn't look right to me. I'm assuming this is the augmented matrix that represents your system. If so, the first row in the matrix should be:
1 0 -1 0 λ

You have
1 0 1 -1 λ

4. May 30, 2013

### Felafel

oops, yes, it should be as you said, but I think it's just a typo, because i've solved it on a piece of paper, so the rest of the calculation should be right and done on the correct matrix

5. May 30, 2013

### Felafel

and on the right side of the third equation there is a λ, yes

6. May 30, 2013

### HallsofIvy

If you mean "is $\lambda= 0$ the correct answer to 'for what values is the system homogeneous,'" then, by the definition of "homogenous system", yes.

7. May 31, 2013

### Felafel

thank you!
and was it correct to use rouchè-capelli's thoerem?