The work considers the set $Λ_n^k$ of all $n\times n$ binary matrices having the same number of $k$ units in each row and each column. The article specifically focuses on the matrices whose rows and columns are sorted lexicographically. We examine some particular cases and special properties of this matrices. Finally, we demonstrate the relationship between the Fibonacci numbers and the cardinality of two classes of $Λ_n^k$-matrices with lexicographically sorted rows and columns.