Expert: Scott A Wilson - 10/6/2009

Question
convert the matrix to echelon form:

1  0  1 -2
2  1  3 -4
-1 0 -1  2
4 -1  1 -1
and find the rank of the matrix.
I understand what echelon form should look like but i dont understand the process of row reduction to achieve the correct form

Answer
To convert a matrix to row echelon form, never multiply a row by anything to replace that row.
Only add multiples of other rows to that row.

R1, R2, R3, R4 = four rows of the matrix.
Leave R1 alone.  Change R2 to R2-2(R1), R3 to R3+R1, and R4 to R4-4(R1).
This gives us
R1 = 1  0  1  2
R2 = 0  1  1  0
R3 = 0  0  0  0
R4 = 0 -1 -3  7

Since R3 is 0's all the way across, the determinant of the matrix is 0.
However, we can add R2 to R4, giving
R1 = 1  0  1  2
R2 = 0  1  1  0
R3 = 0  0  0  0
R4 = 0  0 -2  7

Switching R3 and R4 would make the determinant negative of what is was,
but the determinant is 0 anyway, so the row-echelon form is
R1 = 1  0  1  2
R2 = 0  1  1  0
R3 = 0  0 -2  7
R4 = 0  0  0  0