Hi all.
I cant seem to solve this problem in my math homework:
Solve this matrix using three different methods: Row-reduction, inverse matrix, Cramers rule (using determinants).
The matrix is the following:
[6 4][x]=[4]
[4 3][y] [6]
Solving it using Cramers rule works fine, and I get x=-6 and y=10
Multiplication with the inverse matrix also yields the same solution.
However, when I try to sole it using Row-reduction, I cant make it work:
Starting with the augmented matrix:
[6 4 4] ~~~~~~ 4 3 6 ~~~~~ 4 3 6 ~~~~~~~~~ 0 -2 -8 ~~~~~~ 1 1/4 -7/2
[4 3 6] ~~~~~~ 6 4 4 ~~~~~ 1 1/4 -7/2 ~~~~~ 1 1/4 -7/2 ~~~~~ 0 -1 -4
~1 1 -39/2 ~~~~1 1 -39/2 ~~~~ 0 -1 4 ~~~~~1 2 -47/2
~0 1 -4 ~~~~ 1 2 -47/2 ~~~~ 1 2 -47/2 ~~~~ 0 1 -4 => y=-4 ≠ y=10
Does anyone see what I'm doing wrong? Any help is highly appreciated!
You know the three elementary row operations you're allowed to perform?
Try to post your attempt again, explaining which row operation you are using for each step.
It would be helpful if you used [_code] [/code] tags (remove the _)
Try to post your attempt again, explaining which row operation you are using for each step.
It would be helpful if you used [_code] [/code] tags (remove the _)
QUOTE (beerformyhorses+Sep 29 2008, 03:25 PM)
[6 4 4] ~~~~~~ 4 3 6 ~~~~~ 4 3 6 ~~~~~~~~~ 0 -2 -8 ~~~~~~ 1 1/4 -7/2
[4 3 6] ~~~~~~ 6 4 4 ~~~~~ 1 1/4 -7/2 ~~~~~ 1 1/4 -7/2 ~~~~~ 0 -1 -4
In between your 2nd and 3rd steps, what have you done? were you dividing the 2nd row by 6? Because 4/6 ≠ 1/4 or -7/2.
[4 3 6] ~~~~~~ 6 4 4 ~~~~~ 1 1/4 -7/2 ~~~~~ 1 1/4 -7/2 ~~~~~ 0 -1 -4
In between your 2nd and 3rd steps, what have you done? were you dividing the 2nd row by 6? Because 4/6 ≠ 1/4 or -7/2.
I feel kinda stupid, but heres a more understandable(hopefully) version:
Starting with the augmented matrix:
6 4 4
4 3 6 -subtract 5/4 of lower from Upper
1 1/4 -7/2
4 3 6 -subtract (-4)*upper from lower
0 2 -8
1 1/4 -7/2 -divide upper by 2
0 1 -4
1 1/4 -7/2 - swap places
1 1/4 -7/2
0 1 -4 => y=-4 (right?) ≠ y=10
but Cramers rule tells me that y=10 and x=-6
Starting with the augmented matrix:
CODE
6 4 4
4 3 6 -subtract 5/4 of lower from Upper
1 1/4 -7/2
4 3 6 -subtract (-4)*upper from lower
0 2 -8
1 1/4 -7/2 -divide upper by 2
0 1 -4
1 1/4 -7/2 - swap places
1 1/4 -7/2
0 1 -4 => y=-4 (right?) ≠ y=10
but Cramers rule tells me that y=10 and x=-6
Does anyone see what I'm doing wrong? Any help is highly appreciated!
6-(-7/2 )*4 =20 not -8 and 20/2=10=y
Your steps are a little odd looking; you seem to have missed a swap, and more importantly, all this swapping is pointless. So is your first step.
Basically you want to make your matrix look like:
1 0 -6
0 1 10
In as few steps as possible.
Start with
6 4 4
4 3 6
The quickest way to get the 1 on the top left is just to divide the top row by 6 is it not?
1 2/3 2/3
4 3 6
Now to get the zero on the bottom left just multiply the top row by 4 and subtract it from the bottom:
1 2/3 2/3
0 3-8/3 6-8/3
1 2/3 2/3
0 1/3 10/3
Now to get the zero in the top middle, multiply the bottom row by 2 and subtract it from the top:
1 0 -6
0 1/3 10/3
The just multiply the bottom row by 3 and bingo:
1 0 -6
0 1 10
Your steps are a little odd looking; you seem to have missed a swap, and more importantly, all this swapping is pointless. So is your first step.
Basically you want to make your matrix look like:
1 0 -6
0 1 10
In as few steps as possible.
Start with
6 4 4
4 3 6
The quickest way to get the 1 on the top left is just to divide the top row by 6 is it not?
1 2/3 2/3
4 3 6
Now to get the zero on the bottom left just multiply the top row by 4 and subtract it from the bottom:
1 2/3 2/3
0 3-8/3 6-8/3
1 2/3 2/3
0 1/3 10/3
Now to get the zero in the top middle, multiply the bottom row by 2 and subtract it from the top:
1 0 -6
0 1/3 10/3
The just multiply the bottom row by 3 and bingo:
1 0 -6
0 1 10
He beat me to it!
Yep, in your second line where you say you are doing "-subtract (-4)*upper from lower", you are actually trying to do "-subtract (+4)*upper from lower" AND swap the rows. You just got confused and messed up the one gabba showed you.
Otherwise looks like you've got it cracked!
EDIT (after gabba's edit):
There are certainly simpler ways to get to the final matrix, but it looked to me like you were using a set method which should work for any situation. If you're comfortable with it, stick with it. Once you're really comfortable with matrices, you should start seeing the steps gabba showed you naturally. All I'm saying is don't panic that you took a slightly long way round!
Yep, in your second line where you say you are doing "-subtract (-4)*upper from lower", you are actually trying to do "-subtract (+4)*upper from lower" AND swap the rows. You just got confused and messed up the one gabba showed you.
Otherwise looks like you've got it cracked!
EDIT (after gabba's edit):
There are certainly simpler ways to get to the final matrix, but it looked to me like you were using a set method which should work for any situation. If you're comfortable with it, stick with it. Once you're really comfortable with matrices, you should start seeing the steps gabba showed you naturally. All I'm saying is don't panic that you took a slightly long way round!
Thanks a lot guys!
Like so often, the answer seems straightforward and easy when you just see it.. I've spent more time on that one than I'd like to admit ;D
Like so often, the answer seems straightforward and easy when you just see it.. I've spent more time on that one than I'd like to admit ;D
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