Weight properties of primitive matrices

For nonnegative $n\times n$ matrices ($n>2$), the results of researching the dependence of matrix primitivity on weight (quantity of positive elements) are presented, namely: 1) any matrix of a weight $k\le n$ is not primitive; 2) for $k=n+1,\dots,n^2-n+1$, there are both a not primitive matrix with weight $k$ and a primitive matrix with weight $k$ and exponent $\gamma $ where $n+2\lfloor\sqrt{2(n-1)}\rfloor\le\gamma+k\le n^2-n+3$; 3) any matrix with weight $k=n^2-n+2,\dots,n^2-1$ is primitive and its exponent $\gamma=2$. It is shown that, for some primitive matrices, the weight is not monotonically non-decreasing function of its degree.