$A = \begin{pmatrix} 0.9 & 0.1 \\ 0 & 0.9 \end{pmatrix}$ (diagonal entries less than 1)

$A = \begin{pmatrix} 0.5 & 0.4 \\ 0.4 & 0.5 \end{pmatrix}$ (eigenvalues 0.9 and 0.1)

$A = \begin{pmatrix} 1 & 0.5 \\ 0 & 0.9 \end{pmatrix}$ (trace less than 2)

$A = \begin{pmatrix} 0.3 & 0.5 \\ 0.4 & 0.2 \end{pmatrix}$ (all entries less than 1)