Given the following equations , solve using gauss method " Reduced Row Echelon Form ":

x1 + 2x2 + 3x3 + 2x4 = -1

-x1 -2x2 -2x3 + x4 = 2

2x1 + 4x2 + 8x3 + 12x4 = 4

this is my solution basically , the steps are in order!:

R2 + R1
R3 + (-2)R1
R3 + (-2)R2
R1 + R2(-3)
1/2 R3
R2 + (-3)R3
R1 + R3(7)

And finally got:
1 2 0 0 10
0 0 1 0 -5
0 0 0 1 2

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from what I understand the bottom might not be all 0s , unless there isn't a leading 1

is my solution correct ? and should every column get a leading one ? or what ?

Question

Given the following equations , solve using gauss method " Reduced Row Echelon Form ":

x1 + 2x2 + 3x3 + 2x4 = -1

-x1 -2x2 -2x3 + x4 = 2

2x1 + 4x2 + 8x3 + 12x4 = 4

this is my solution basically , the steps are in order!:

R2 + R1
R3 + (-2)R1
R3 + (-2)R2
R1 + R2(-3)
1/2 R3
R2 + (-3)R3
R1 + R3(7)

And finally got:
1 2 0 0 10
0 0 1 0 -5
0 0 0 1 2

---

from what I understand the bottom might not be all 0s , unless there isn't a leading 1

is my solution correct ? and should every column get a leading one ? or what ?