Extensions of some matrix inequalities related to trace and partial traces

We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends of Lin [Czech. Math. J. 66 (2016)] and improves Kuai [Linear Multilinear Algebra (2018)]. Moreover, we provide unified treatment Ando [ILAS Conference (2014)] recent Li, Liu Huang [Operators Matrices 15 (2021)]. Furthermore, also extend some inequalities involving larger class matrices whose numerical ranges are contained in sector. In addition, extensions on trace $2\times 2$ included.