need help with hw...
Use Gauss-Jordan to solve
3x – 2y + z = 6
3x + y – z = -4
-x + 2y – 2z = -8
Start by transcribing the equations to a 3x4 matrix
3 -2 1 6
3 1 -1 -4
-1 2 -2 -8
The first step is to reduce each column to zero (except the diagonal element and above) by linear combination of rows. I'll do the first one:
3 -2 1 6
0 3 -2 -10 (row 2 - row 1)
0 4 -5 -18 (3*row 3 + row 1)
Continue:
3 -2 1 6
0 3 -2 -10
0 0 7 14 (4*row 2 - 3*row 3)
The last row can be simplified:
3 -2 1 6
0 3 -2 -10
0 0 1 2
So z=2, substitute z=2 in row 2 to eliminate elements above the diagonal
3 -2 0 4 (row 1 - row 3)
0 3 0 -6 (row 2 + 2*row 3)
0 0 1 2
Simplify row 2:
3 -2 0 4
0 1 0 -2 (row 2/3)
0 0 1 2
Now eliminate column 2 from row 1
3 0 0 0 (row 1 + 2*row 2)
0 1 0 -2
0 0 1 2
So
x=0
y=-2
z=2