Why Is My Determinant Calculation Using Upper Triangular Form Incorrect?

[Dethrone]

I am trying to find the determinant of the following via upper triangular form:

$$\left[\begin{array}{c}-1 & -1 & 1 & 0 \\ 2 & 1 & 1 & 3 \\ 0 & 1 & 1 & 2 \\ 1 & 3 & -1 & 2 \end{array}\right]$$Using row reduction to bring it to upper triangular matrix:

$$\left[\begin{array}{c}-1 & -1 & 1 & 0 \\ 0 & -1 & 3 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1/2 \end{array}\right]$$Proposition from my professor's notes (modified a bit as we know the determinant exists and is unique):

Let $\text{det}_n: \Bbb{R}^{nn}\longmapsto\Bbb{R}$ be the determinant function and $U=[u_{ij}]\in\Bbb{R}^{nn}$ by an upper-triangular matrix. Then
$$\text{det}_n(U)=\prod_{k=1}^nu_{kk}$$.

Applying this formula I get $(-1)(-1)(1)(1/2)=1/2$ as my determinant, whereas the actual determinant is $2$. Why is this wrong? Furthermore, I can see that I can multiply the last row by any constant, therefore, the determinant can really be any real number. What is wrong with my application of the proposition?

 

[Dethrone]

On second though, could this be the reason:

Of the three elementary row operations, only the third type where you multiply one row by a scalar and add it to another row doesn't change the determinant value. In other words, I should either keep track of the row operations I use, or avoid the other two row operations altogether?

 

[Nono713]

That's right, the determinant is not necessarily preserved by row operations. You should keep track of the determinant as you work through the row reduction, then quickly calculate the determinant of the upper-triangular matrix, and then work backwards to find the determinant of the original matrix. It's still fast and easy, especially since row reductions can only either negate or multiply the determinant by a constant factor, so the correction factor is always of the form $(-1)^n s$ where $n$ is the number of times you swap rows and $s$ is the product of all scalar row multiplications you perform (that's multiplying a row by a scalar, not adding a multiple of a row to another, since that does not affect the determinant).

 

[Dethrone]

Hi Bacterius,

Thanks very much for clearing this up for me, as always! :D