About the minimal primitive matrices

A quadratic Boolean matrix $A$ is called a primitive matrix if some its degree does not contain 0's. A primitive matrix is called a minimal primitive matrix if it becomes non-primitive matrix after replacing any one 1 in it by 0. The height of a primitive matrix is defined as the least Hamming's distance between it and a minimal primitive matrix. In the paper, properties of minimal primitive matrices are studied. The amount of minimal primitive matrices of order $n$ is estimated. An algorithm for estimating the height of a primitive matrix is proposed.