This paper presents the survey of results related to the bounded linear operator matrices on Banach spaces (specially on Hilbert spaces) and the results in Banach algebras, which are related to $2\times2$ operator matrices. We observe bounded linear operators represented in a matrix form, as well as the elements of Banach algebra in the similar form. Also, we study Fredholm properties and generalized invertibility of such operators. The limited space did not allow us to present all related results, so we believe that the enlarged paper will be the future project. The paper contains some new results on generalized and hypergeneralized projections.