Matrices: Find x,y,z, and w by the GAUSE- JORDAN method.?

Question

5x+3y+z+5w=-2
3x+2y+2z+4w=2
2x+4y+3z-3w=-11
4x-3y-2z+2w=-3

Answer 1

Well Aq = b. where A is the matrix of coefficients, q is [x; y; z; w], and b is the answer [-2; 2; -11; -3] So we are looking for the Rx = d, that corresponds to the original equation. The aumented matrix is
5 3 1 5 -2
3 2 2 4 2
2 4 3 -3 -11
4 -3 -2 2 -3
The first step is to create a 1 in 1,1 so we divide Row1 by 5.
Then subtract 3xRow1 from row 2, 2xRow1 from Row3, 4xRow1 from Row4
This is the result
1 0.6 0.2 1 -0.4
0 0.2 1.4 1 3.2
0 2.8 2.6 -5 -10
0 -5.4 -2.8 -2 -1.4
Continue down and to the right with row operations in the same vein until the matrix is in row echelon form then go up and to the left in order to create 0s in all the non-step positions and get the matrix in reduced row echelon form.

The reduced row echlon form of the augmented matrix [A b] is
1 0 0 0 -2
0 1 0 0 -1
0 0 1 0 1
0 0 0 1 2
(Heh look at the nice numbers. You got this out of a textbook didn't ya?)
This was of course gotten by the Gauss-Jordan Method.
So now it's just a matter of reading off the coefficients
1q1 + 0q2 + 0q3 + 0q4 = -2
0q1 + 1q2 + 0q3 + 0q4 = -1
0q1 + 0q2 + 1q3 + 0q4 = 1
0q1 + 0q2 + 0q3 + 1q4 = 2

So q = [-2; -1; 1; 2] and x = -2, y = -1, z = 1, w =2.

If you want all the row operations, well there are an awful lot of them and they're all simple and straightforward. It's simply a matter of carefully working through the algorithm, so you are better off working them out for yourself. You'll probably get them before I can type them all in.