Let $\mathbf{A} = \begin{pmatrix} 1 & 4 \\ 2 & b \end{pmatrix}$. If $\mathbf{A}^2 = 5 \mathbf{A} - 6 \mathbf{I}$, find the value of $b$. Let's think step by step and output the final answer within \boxed{}.
user