I've been asked to: use determinant properties to evaluate the 5 x 5 determinant given below.

$\displaystyle \begin{bmatrix}1 & 3 & 1 & -6 & 8 \\ -1 & 2 & -1 & 6 & -8 \\ 0 & 0 & 4 & 8 & -4 \\ 2 & 6 & 0 & -16 & 18 \\ 0 & 0 & 1 & 2 & 2\end{bmatrix}$

I've only just started learning matrix algebra, but I was aiming to reduce this matrix to a triangle matrix so I could use the main diagonal to calculate the determinant. I have a couple of questions though:

1 - Does this method count as "using determinant properties"? Or should I be doing it differently in order to answer the question correctly?

2 - Is my working below (and resulting determinant) correct?

WORKING STEPS:

1 - Add the first row to the second.

$\displaystyle \begin{bmatrix}1 & 3 & 1 & -6 & 8 \\ 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 4 & 8 & -4 \\ 2 & 6 & 0 & -16 & 18 \\ 0 & 0 & 1 & 2 & 2\end{bmatrix}$

2 - Subtract 2x the first row from the fourth.

$\displaystyle \begin{bmatrix}1 & 3 & 1 & -6 & 8 \\ 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 4 & 8 & -4 \\ 0 & 0 & -2 & -28 & 2 \\ 0 & 0 & 1 & 2 & 2\end{bmatrix}$

3 - Add 2x the fifth row to the fourth.

$\displaystyle \begin{bmatrix}1 & 3 & 1 & -6 & 8 \\ 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 4 & 8 & -4 \\ 0 & 0 & 0 & -24 & 6 \\ 0 & 0 & 1 & 2 & 2\end{bmatrix}$

4 - Subtract 1/4x the third row from the fifth.

$\displaystyle \begin{bmatrix}1 & 3 & 1 & -6 & 8 \\ 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 4 & 8 & -4 \\ 0 & 0 & 0 & -24 & 6 \\ 0 & 0 & 0 & 0 & 3\end{bmatrix}$

Because this is now a triangle matrix, the determinant should just be equal to the multiple of the main diagonal right? i.e. 1 x 5 x 4 x -24 x 3 = -1440

I have a strong feeling that this is wrong, and possibly that the 0 5 0 0 0 row is more significant than I realise. Any comments/help would be greatly appreciated.