Is there any way of showing a matrix is regular without multiplying it by itself until all entries are positive?
"multiplying by itself until all entries are positive"?? Who told you this way you can find out whether a matrix is regular? Take, for example, the real 2x2 matrix whose all entries are 1: all its elements are positive but it is not regular.
A nice way is calculating its determinant: the matrix is regular iff its the determinant is non zero.
Another way: bring your matrix to echelon form though elementary operations on its rows (or columns): then the matrix is regular iff at no step a row (or column) becomes all zeroes.
There are severaa other characterizations of regular matrices but the above two are, perhaps, of the more basic ones.