Let \(n\), \(r\) be nonnegative integers with \(r< n\). An \(n\times n\) matrix \(A\) is called \(r\)-partly de(composable) if it contains a \(k\times l\) zero submatrix with \(k+ l= n -r+ 1\). If \(A\) is not \(r\)-partly dec, it is called \(r\)-indec (a \(0\)-indec matrix is call a Hall matrix). If \(A\) is \(r\)-indec but, for every non-zero element of \(A\), the matrix \(A\) with that element replaced by \(0\) is \(r\)-partly dec, then \(A\) is called \(r\)-nearly dec. In this paper the authors derive numerical and enumerative results on \(r\)-nearly dec 0-1 matrices as well as bounds on the convergence index of \(r\)-indec matrices, in particular for the adjacency matrices of primitive Cayley digraphs and circulant matrices.