solving systems of equations with matrices

M

MWoods

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Sep 4, 2010
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I have a system of equations:
w + x + 5y - z = 0
x + y - 2z = 0
x + 2y - 5z = 0

The problem is that when I manipulate to find the Identity matrix ( 1s on the major diagonal ) I don't get a "z" quantity because the matrix is a 3X5. Can you tell me what I need to do when solving a system like this? Thank you.

Mwoods
 


Please show us the echelon form that you got.

You can type matrices like this example:

[1, 0, 3, -4, 10]
[0, 1 , 4, -7, 8]
[1, 1, -1, 7, 2]

Cheers!

 
here it is:

[1, 0, 0, -1/3 | 0]
[0, 1, 0, -7/3 | 0]
[0, 0, 1, 1/3 | 0]

that is as far as I got.
 
\(\displaystyle \begin{bmatrix}1&1&5&-1&0\\0&1&1&-2&0\\0&1&2&-5&0\end{bmatrix}\)

\(\displaystyle -R_{2}+R_{3}\rightarrow R_{3}\):

\(\displaystyle \begin{bmatrix}1&1&5&-1&0\\0&1&1&-2&0\\0&0&1&-3&0\end{bmatrix}\)

\(\displaystyle -R_{3}+R_{2}\rightarrow R_{2}\):

\(\displaystyle \begin{bmatrix}1&1&5&-1&0\\0&1&0&1&0\\0&0&1&-3&0\end{bmatrix}\)

\(\displaystyle -R_{2}+R_{1}\rightarrow R_{1}\):

\(\displaystyle \begin{bmatrix}1&0&5&-2&0\\0&1&0&1&0\\0&0&1&-3&0\end{bmatrix}\)

\(\displaystyle -5R_{3}+R_{1}\rightarrow R_{1}\)

\(\displaystyle \begin{bmatrix}1&0&0&13&0\\0&1&0&1&0\\0&0&1&-3&0\end{bmatrix}\)

So, we have it whittled down and we get:

\(\displaystyle \begin{vmatrix}w=-13z\\x=-z\\y=3z\end{vmatrix}\)

Unless you may want to write it another way.
 


I also get:

[1, 0, 0, 13, 0]
[0, 1, 0, 1, 0]
[0, 0, 1, -3, 0]

z is a parameter for the solution set.

In other words, z can be any Real number and w, x, y are all functions of z.

It's standard to include z = z, as the first line of the solution posted by galactus.

 
I was looking at your answer and realized that I totally messed up the system of equations. Sorry. It should be:

w + x + 5y -z = 0
x + y - 2z = 0
w + 2x - 5z = 0

that was totally my fault. But, the real help was how you explained that w,x,y are functions of z. taht's the missing link for me. I can do the problem now. Thanks for the great help.

Double Cheers!! to all contributors
 
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