Construction of a non-negative matrix

Question

I am designing an algorithm, in which an intermediate step is to create a $n\times n$ invertible non-negative matrix. The idea that I thought was to choose a random generator $P$ to generate $n^2$ points and then make a matrix $A$ by reshaping. Then do a $A^TA$ to make it non-negative and then $B=I+A^TA$ to make it invertible. But the problem is that since B is a symmetric matrix that will be a drawback for my algorithm which I don't need. Can somebody suggest any readings or methods that construct the non-negative matrix? Edit: B should be a matrix whose entries are uncorrelated. The matrix $B$ should be sparse