Conditions for affine transform on $\mathbb{R}^3$ to be a contraction

Question

I am reading about affine transformations used in fractal image compression, they are of the form:

$$ w \begin{pmatrix} x \\ y \\z \end{pmatrix} = \begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & s \\ \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} + \begin{pmatrix} e \\ f \\ o \\ \end{pmatrix}$$

Let $$ A = \begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & s \\ \end{pmatrix} $$

$w$ must be a contraction: spatial contraction (acting on the plane: $x$- and $y$-coordinates) and also a contraction in the $z$-direction. Some of the conditions for $w$ to be a contraction (that I found in different sources) are:

1. $\| A \|_{2} < 1$
2. $ |s| < 1$ and $\left\| \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \right\|_{2} < 1$
3. $ |\det (A)| < 1$ (as a condition for spatial contraction)

What are the right conditions for spatial contractivity and z-contractivity ? Thanks in advance.