Suppose \([A_{ij}]\) is an \(n\times n\) operator matrix, where each \(A_{ij}\) is a bounded linear operator on a complex Hilbert space \(\mathcal {H}\) . Among other inequalities, it is shown that \(w([A_{ij}]) \le w([a_{ij}]),\) where \([a_{ij}]\) is an \(n\times n\) matrix with \(\begin{aligned} a_{ij}={\left\{ \begin{array}{ll} w(A_{ii}) & \text {if } i=j,\\ \underset{0\le t \le 1}{\min }\ \left\| |A_{ij}|^{2t} + |A_{ji}^*|^{2t} \right\| ^{1/2} \left\| |A_{ij}^*|^{2(1-t)}+ |A_{ji}|^{2(1-t)} \right\| ^{1/2} & \text {if } i< j,\\ 0 & \text {if } i> j. \end{array}\right. } \end{aligned}\) This numerical radius bound refines a well known bound by Abu-Omar and Kittaneh [Linear Algebra Appl. 468 (2015), 18–26]. We use these estimates to derive several numerical radius inequalities and equalities for \(2\times 2\) operator matrices. Applying these inequalities, we also deduce several numerical radius bounds for a bounded linear operator, the product of two operators and the commutator of operators. In particular, it is shown that \(\begin{aligned} w(A) \le \underset{0\le t \le 1}{\min }\ \left( \frac{1}{2} \Vert A\Vert ^t \left\| |A|^{1-t}+|A^*|^{1-t} \right\| \right) , \end{aligned}\) where A is a bounded linear operator on \(\mathcal {H}\) . This bound refines as well as generalizes the well known bounds.
