System of Linear Equations (Solve Using Matrices)

[TheWrathOfMath]

Please see attached file which includes a set of linear equations.

1. For what values of alpha and beta the system has one solution?
2. For what values of alpha and beta the system has no solutions?
3. For what values of alpha and beta the system has infinitely many solutions?
4. For every case in which the system has infinitely many solutions, find the general solution of the system.

 

[blamocur]

READ BEFORE POSTING

Welcome to FreeMathHelp.com! Please take the time to read the following before you make your first post. It will help you to get your math questions answered promptly and in the most helpful manner. A summary is available here, but please read the complete guidelines below at your earliest...

www.freemathhelp.com

 

[topsquark]

What have you been able to do with this?

-Dan

 

[TheWrathOfMath]

What have you been able to do with this?

-Dan

I probably first need to write the equations in the form of a matrix and change it into the reduced row echelon form, right?

And then I will need to compute the rank of the matrix by setting the leading entry in every row equal to zero?

And then use it to figure out how many solutions the non-homogenous system of linear equations has?

Do I need to change the matrix so that it is in the row echelon form or the reduced row echelon form?

 

[blamocur]

Do I need to change the matrix so that it is in the row echelon form or the reduced row echelon form?

Whatever works

 

[TheWrathOfMath]

Whatever works

What about the other questions I asked in the above comments?

 

[Dr.Peterson]

What about the other questions I asked in the above comments?

Do what you suggested and see if it works! (It probably will, if you carry it out appropriately.) I think that's what everyone is waiting to see.

 

[TheWrathOfMath]

Do what you suggested and see if it works! (It probably will, if you carry it out appropriately.) I think that's what everyone is waiting to see.

Whatever works

What have you been able to do with this?

-Dan

Does this look okay so far?
I changed the matrix into row echelon form.

If this is correct, do I now need to compute the rank of the matrix A and the augmented matrix (A|b) by setting the leading entry -alpha-beta equal to zero?

 

[TheWrathOfMath]

What have you been able to do with this?

-Dan

this look okay so far?
I changed the matrix into row echelon form.

If this is correct, do I now need to compute the rank of the matrix A and the augmented matrix (A|b) by setting the leading entry -alpha-beta equal to zero?

Do what you suggested and see if it works! (It probably will, if you carry it out appropriately.) I think that's what everyone is waiting to see.

Whatever works

For instance, for "no solution" I will probably need to set -alpha-beta=0 and -alpha(3+beta)=/=0 so that r(A|b)>r(A), right?

 

[blamocur]

For instance, for "no solution" I will probably need to set -alpha-beta=0 and -alpha(3+beta)=/=0 so that r(A|b)>r(A), right?

I agree. BTW, what happens when [imath]\alpha+\beta = 0[/imath] and [imath]\alpha(3+\beta) = 0[/imath] ?

 

[TheWrathOfMath]

I agree. BTW, what happens when [imath]\alpha+\beta = 0[/imath] and [imath]\alpha(3+\beta) = 0[/imath] ?

There is no solution when alpha= -beta and beta=/=0, beta=/= -3, right?

 

[TheWrathOfMath]

I agree. BTW, what happens when [imath]\alpha+\beta = 0[/imath] and [imath]\alpha(3+\beta) = 0[/imath] ?

I do not know if I did this correctly, since I am not supposed to receive a value of alpha in terms of beta.
I think I need to find numerical values for alpha and beta.
Perhaps I should have computed the row echelon form differently?

 

[blamocur]

Does this look okay so far?
I changed the matrix into row echelon form.

If this is correct, do I now need to compute the rank of the matrix A and the augmented matrix (A|b) by setting the leading entry -alpha-beta equal to zero?

You echelon form looks correct, but more complex than necessary. Why did you need to apply [imath]R_1 \longrightarrow R_1 - \alpha R_2[/imath] ?

 

[blamocur]

I do not know if I did this correctly, since I am not supposed to receive a value of alpha in terms of beta.
I think I need to find numerical values for alpha and beta.
Perhaps I should have computed the row echelon form differently?

After the reduction you have [imath](\alpha+\beta) w = \alpha\beta + 3\alpha[/imath]. What do you think about this equation : [imath]0\cdot w = 0[/imath] ?

 

[blamocur]

There is no solution when alpha= -beta and beta=/=0, beta=/= -3, right?

I agree.

 

[TheWrathOfMath]

I agree.

But I got alpha in terms of beta.
Wasn't I supposed to get it in terms of a number?
Or not necessarily?

 

[blamocur]

But I got alpha in terms of beta.
Wasn't I supposed to get it in terms of a number?
Or not necessarily?

I haven't noticed any requirements of expressing alpha in terms of beta.

 

[blamocur]

I do not understand what you mean.

What can you say about the solutions when [imath]\alpha+\beta = 0[/imath] and [imath]\alpha\beta+3\alpha=0[/imath] ?

 

[TheWrathOfMath]

What can you say about the solutions when [imath]\alpha+\beta = 0[/imath] and [imath]\alpha\beta+3\alpha=0[/imath] ?

I don't know anymore.
The more I try to understand it, the more confused I get.
Never mind.
Thank you for trying to help.

 

[blamocur]

+

I don't know anymore.
The more I try to understand it, the more confused I get.
Never mind.
Thank you for trying to help.

If both parts are zero then you get [imath]0\cdot w = 0[/imath]. Which values of [imath]w[/imath] would fit this equation?