The paper describes a way of representing of isotone Boolean functions over $B_2$. A class of binary matrices is used in order to characterize the entire class of isotone Boolean functions. It is proved that the matrix corresponding to an isotone Boolean function is an $M$-matrix and, conversely, that each $M$-matrix corresponds to an isotone Boolean function. The $M$-matrix is defined as a binary matrix of such a kind that any two of its rows are noncomparable vectors. The results are applied for counting the functions belonging to some subclasses of the class of isotone Boolean functions.