Thread: 3x3 Matrix Problem - Gaussian Elimination

Hello, this is the begining to my Pre-Calc class and I'm having issues with a 3x3 Matrix.
The problem is as follows:

8r-s+t=34
2r+2s-3t=25
r-3s+2t=-10

This is what I have so far:

[8 -1 1 | 34]
[2 2 -3 | 25]
[1 -3 2 |-10]

R1<->R3
[1 -3 2 | -10]
[2 2 -3 | 25]
[8 -1 1 | 34]

-2R1+R2

[1 -3 2 |-10]
[0 -4 6 |-50]
[8 -1 1 | 34]

-8R1+R3

[1 -3 2| -10]
[0 -4 6| -50]
[0 23 -8|-272]

I'm stuck here. I know that I need to get R2C2 to a 1 and assuming I'm correct its a problem of simple math, but other than use a calculator to figure it out by trial and error how do I find out a simple method to get the one? I realize this is a simple case of math, but to be frank I'm in an online class and I don't get the great base of Math most get.

Thanks for any advice anyone can give.

Originally Posted by MauiCormac

Hello, this is the begining to my Pre-Calc class and I'm having issues with a 3x3 Matrix.
The problem is as follows:

8r-s+t=34
2r+2s-3t=25
r-3s+2t=-10

This is what I have so far:

[8 -1 1 | 34]
[2 2 -3 | 25]
[1 -3 2 |-10]

R1<->R3
[1 -3 2 | -10]
[2 2 -3 | 25]
[8 -1 1 | 34]

-2R1+R2

[1 -3 2 |-10]
[0 -4 6 |-50]
[8 -1 1 | 34]

-8R1+R3

[1 -3 2| -10]
[0 -4 6| -50]
[0 23 -8|-272]

I'm stuck here. I know that I need to get R2C2 to a 1 and assuming I'm correct its a problem of simple math, but other than use a calculator to figure it out by trial and error how do I find out a simple method to get the one? I realize this is a simple case of math, but to be frank I'm in an online class and I don't get the great base of Math most get.

Thanks for any advice anyone can give.

You can multiply R2 by -1/4.

There are three "row operations":
1) Swap two rows.
2) Multiply or divide any row by a fixed number.
3) Add or subtract a multiple of a row from another row.

Any time you have a "pivot" number (the number leading the row you are working with which should be on the diagonal) that is not 1 (or 0), you can make it 1 by dividing the entire row by that number.

(If it is 0, look for a lower row in which the leading number is not 0 and swap. If the leading numbers in all lower rows is 0, the matrix is singular and you cannot continue.)

Ahh that makes sense. Ok thanks! I'm sure I'll have more questions here the further I get into. This website is a great resource.

So heres that question I was talking about.

I multiply R2 by -1/4 which gives me;

[1 -3 2 |-10 ]
[0 1 -3/2 |25/2 ]
[0 23 -8 |-272 ]

From here I would do -23R2+R3 correct?

Then I work back through substitution? I seem to have made an error somewhere assuming my theory is correct. I'm not sure why I'm struggling so much, the concept seems easy enough to me.

Nevermind. I found I made a big error in the beginning and now I'm just totally bumfuzzled. I think I'll sleep on it and come back unfrustrated.

Originally Posted by MauiCormac

So heres that question I was talking about.

I multiply R2 by -1/4 which gives me;

[1 -3 2 |-10 ]
[0 1 -3/2 |25/2 ]
[0 23 -8 |-272 ]

From here I would do -23R2+R3 correct?

Then I work back through substitution? I seem to have made an error somewhere assuming my theory is correct. I'm not sure why I'm struggling so much, the concept seems easy enough to me.

Looking more carefully at your work, seems you made an error early on

[1 -3 2 | -10]
[2 2 -3 | 25]
[8 -1 1 | 34]

-2R1+R2

[1 -3 2 |-10]
[0 -4 6 |-50]
[8 -1 1 | 34]

I get

[1 -3 2 | -10]
[0 8 -7 | 45]
[8 -1 1 | 34]

Edit: Didn't see your latest post.

Ok so I got to that point after I found my early error.

So now I go -8R1+R3 which gives me

[1 -3 2 | -10 ]
[0 8 -7 | 45 ]
[0 23 -15 | 114 ]

Where do I go from here? I want to eliminate 8. Which I know I can do with 1/8 which will give me the desired 1 but fractions. Is there an easier way to go from here?

Originally Posted by MauiCormac

Ok so I got to that point after I found my early error.

So now I go -8R1+R3 which gives me

[1 -3 2 | -10 ]
[0 8 -7 | 45 ]
[0 23 -15 | 114 ]

Where do I go from here? I want to eliminate 8. Which I know I can do with 1/8 which will give me the desired 1 but fractions. Is there an easier way to go from here?

I think multiplying R2 by 1/8 will be the easiest way; not all problems are designed so that all steps involve nice integers. If you don't multiply R2 by 1/8, then your next step would probably be (-23/8)R2 + R3 which is awkward.

Ok, I finally got it. Solution (5,3,-3). I think I had it basically right all a long. I just had SO many small errors and problems with my housekeeping of everything that I just got frustrated and started missing stuff.

Thanks for the help undefined. Really appreciate it.