Some Involutory Pascal Matrices Can Make More Involutory Pascal Matrices

There is an intuitive way to flip signs within a lower triangular matrix n×n matrix whose entries form the first n – 1 rows of Pascal's triangle to cause the matrix to be its own multiplicative inverse. For any given positive integer n, we find how many unintuitive choices of flipped signs will also cause the n × n matrix to be its own multiplicative inverse by first seeing how such matrices may or may not fit into larger matrices from the same family.