Determinant of a $4\times4$ matrice with one unknown? I have to calculate the determinant of this matrice. I want to use the rule of sarrus, but this does only work with a $3\times3$ matrice: $$ A= \begin{bmatrix} 1 & -2 & -6 & u \\\ -3 & 1 & 2 & -5 \\\ 4 & 0 & -4 & 3 \\\ 6 & 0 & 1 & 8 \\\ \end{bmatrix} $$ $|A|=35$ Any idea how to solve this?

$$A=\begin{pmatrix}1 & -2 & -6 & u \\\\-3 & 1 & 2 & -5 \\\4 & 0 & -4 & 3 \\\6 & 0 & 1 & 8 \\\\\end{pmatrix}\stackrel{ 3R_1+R_2}{\stackrel{-4R_1+R_3}{\stackrel{-6R_1+R_4}\longrightarrow}}\begin{pmatrix}1 & -2 & \;\;-6 & \;\;\;u \\\0 & -5 & -16 & \;\;\,3u-5 \\\0 & \;\;\;8 & \;\;\;20 &-4u+ 3 \\\0 & \;\,12 & \;\;\;37 & -6u+8 \\\\\end{pmatrix}$$

Now develop the above wrt the first column and you get a $\,3\times 3\,$ determinant. Compute now directly or repeat the above process.