Advice on calculating the determinant for 3x3 Matrix by inspection

[SubZer0]

Homework Statement

The problem is to calculate the determinant of 3x3 Matrix by using elementary row operations. The matrix is:

A =
[1 0 1]
[0 1 2]
[1 1 0]

Homework Equations

The Attempt at a Solution

By following the properties of determinants, I attempt to get a triangular matrix. The steps I follow are:

R3 = (-1)*R1 + R3
[1 0 1]
[0 1 2]
[0 1 -1]

R3 = (-1)*R3 + R2
[1 0 1]
[0 1 2]
[0 0 3]

Which is now a triangular matrix. To calculate the determinant, it should be a simple matter of multiplying the elements of the diagonal, eg. 1 * 1 * 3 = 3. If I calculate the determinant by cofactor, the determinant is -3.
By following the row operations (adding a multiple of a row), this should not affect the determinant. Where am I going wrong?

Retrospectively, doing a row swap of 1 and 3 would have been easier, but this, theoretically, should work.

 

[Orodruin]

You can only add/subtract other rows to a row. The linear combination you replace the row with must have 1 as the prefactor of the old row. In your case you are doing

R3 = (-1)*R3 + R2

which has -1 as the prefactor. You need to do R3new = R3 - R2. Because you put a minus sign before R3, you are getting a sign error.

 

[SubZer0]

Thanks, Orodruin, this makes perfect sense now. Thanks for the explanation.