Norm attainment of a class of block operator matrices

Given complex numbers $a, b, c$ and a non-negative continuous function $\varphi$ defined on $[0, +\infty)$, consider the $2 \times 2$ matrix $$ M_t = \begin{pmatrix} a & t \\ ct & b\varphi(t) \end{pmatrix}, \quad t \in [0, +\infty). $$ We establish conditions for the strict monotonicity of the norm function $t \mapsto \|M_t\|$. As an application, we characterize the norm attainment of the corresponding block operator matrix $$ T = \begin{pmatrix} aI_H & A \\ cA^* & b\varphi(|A|) \end{pmatrix},$$ where $I_H$ is the identity operator on a Hilbert space $H$ and $A$ is a bounded linear operator from another Hilbert space to $H$.