<s>[INST] Simple determinant calculation 
\begin{vmatrix} 
1 & -2 & 2 \\
1 & -2 & -2\\
2 & -2 & 1 
\end{vmatrix}

It is fairly easy I just want to reassure the steps are correct:
\begin{vmatrix} 
1 & -2 & 2 \\
1 & -2 & -2\\
2 & -2 & 1 
\end{vmatrix}  Taking out $-2^{n}$ when $n$ is the number of rows/columns in this case $n=1$ (the second column)  
$$-2 \begin{vmatrix} 
1 & 1 & 2 \\
1 & 1 & -2\\
2 & 1 & 1 
\end{vmatrix} $$
Reducing column 2 from column 1 as no effect on the determinant
$$-2 \begin{vmatrix} 
0 & 1 & 2 \\
0 & 1 & -2\\
1 & 1 & 1 
\end{vmatrix} $$
Using Laplace expansion on the first column
$$(-2)*1[1*(-2)-2*1]=8.$$
 [/INST] \begin{vmatrix} 
1 & -2 & 2 \\
1 & -2 & -2\\
2 & -2 & 1 
\end{vmatrix}
$$R'_1=R_1-R_2 \space R'_2=R_2-R_3 $$
$$\begin{vmatrix} 
0 & 0 & 4 \\
-1 & 0 & -3\\
2 & -2 & 1 
\end{vmatrix}
=8$$
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