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# Tricky Gaussian Matrix watch

1. I've worked this one out, and know how to apply the gaussian rule.
And the augmented matrix ends up having 0 0 0 on the third row.

I have the question and mark scheme,
I did everything exactly the same, aside from using z = alpha

Can someone please kindly explain why this is done, and if possible link me to a website that explains the theory behind doing this.

Thanks!

http://s1352.photobucket.com/user/An...308ba.jpg.html

Also what is meant by 'most general form of solutions?'
2. (Original post by Sifr)
I've worked this one out, and know how to apply the gaussian rule.
And the augmented matrix ends up having 0 0 0 on the third row.

I have the question and mark scheme,
I did everything exactly the same, aside from using z = alpha

Can someone please kindly explain why this is done, and if possible link me to a website that explains the theory behind doing this.

Thanks!

http://s1352.photobucket.com/user/An...308ba.jpg.html

Also what is meant by 'most general form of solutions?'
Essentially, in the augmented matrix, if you have a row of 4 zeros, then there are infinite solutions to your system of equations. It makes sense to think about it graphically - when you have an equation with 3 variables, it describes a plane. What 4 zeros in the augmented matrix means is that all 3 planes intersect in a line which extends infinitely - infinite solutions.

When you want to find the general set of solutions for this, you introduce a new parameter - doesn't matter what you call it (I normally use , they used in this case). You let one of your variables = this new parameter - here they let z = alpha (which is the new parameter - normally you do let your z variable = the new parameter but it doesn't matter).

You see when you have a system of equations in the augmented matrix? If you know that there are infinite solutions (i.e. 4 zeros in one row) then what you want to do is to eliminate one variable. This lets you write all 3 of your variables in terms of your new parameter. So for example, look at the reduced row echelon form of the matrix in the middle of the page in your link. You have

Consider the first row now, where we've eliminated 'y'. If we let we have that

Now consider the 2nd row => You have that

There's a typo in the last line as you can see when they solved for y in the third line from the bottom, they get

Thus the solution to your system of equations is:

or

(which you should recognise is the equation of a line)

Hope that helped
3. (Original post by Felix Felicis)
Essentially, in the augmented matrix, if you have a row of 4 zeros, then there are infinite solutions to your system of equations. It makes sense to think about it graphically - when you have an equation with 3 variables, it describes a plane. What 4 zeros in the augmented matrix means is that all 3 planes intersect in a line which extends infinitely - infinite solutions.

When you want to find the general set of solutions for this, you introduce a new parameter - doesn't matter what you call it (I normally use , they used in this case). You let one of your variables = this new parameter - here they let z = alpha (which is the new parameter - normally you do let your z variable = the new parameter but it doesn't matter).

You see when you have a system of equations in the augmented matrix? If you know that there are infinite solutions (i.e. 4 zeros in one row) then what you want to do is to eliminate one variable. This lets you write all 3 of your variables in terms of your new parameter. So for example, look at the reduced row echelon form of the matrix in the middle of the page in your link. You have

Consider the first row now, where we've eliminated 'y'. If we let we have that

Now consider the 2nd row => You have that

There's a typo in the last line as you can see when they solved for y in the third line from the bottom, they get

Thus the solution to your system of equations is:

or

(which you should recognise is the equation of a line)

Hope that helped
Thanks so much! That makes a lot more sense now
4. (Original post by Sifr)
I've worked this one out, and know how to apply the gaussian rule.
And the augmented matrix ends up having 0 0 0 on the third row.

I have the question and mark scheme,
I did everything exactly the same, aside from using z = alpha

Can someone please kindly explain why this is done, and if possible link me to a website that explains the theory behind doing this.

Thanks!

http://s1352.photobucket.com/user/An...308ba.jpg.html

Also what is meant by 'most general form of solutions?'
If one of your rows reduces to all zeroes, then you are left with two equations from which you are trying to determine 3 unknowns. This is impossible, though, and the only way to solve it is in terms of a parameter.

Considering the problem geometrically, the two equations you are left with each describe a plane. Two planes cannot meet in one point; they either meet in a line, are parallel, or are the same plane.

We could let any of x, y and z by any parameter we like, really, as long as we are consistent. You want to get x and y in terms of z; or x and z in terms of y; or y and z in terms of x.

A one-parameter solution describes a line. Sometimes you can get a two-parameter solution which describes a plane. This is consistent with the vector equations for lines and planes which have 1 and 2 parameters respectively.

I think by the most general solution they just mean that you don't assign a specific value to your parameter. If then you can obviously choose any value for x, y or z. However, they want the solution to be general, so they used . If you decide , then the solution is not general.

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