Multiply two lower triangular matrices and get another lower triangular matrix. Just try it and you'll see. Take the ith row, multiply it by all the columns in the second matrix, and everything after column i is 0. Also, if both matrices have ones down the main diagonal, the same holds for the product.
Use the previous procedure to invert a lower triangular matrix. If M is lower triangular and c is the ith column of the inverse, we must solve for c, where M*c gives the ith column in the identity matrix. Since M is already triangular, we can solve for the variables directly. Find a column vector c, such that M*c gives the ith column in the identity matrix. The first entry c1 must be 0, along with c2, and c3, and all of them up to but not including ci. Note that ci is the inverse of Mi,i. If M has ones down the main diagonal, so does its inverse.
The lower triangular matrices form a subgroup of the nonsingular matrices, and the lower triangular matrices with determinant 1 form a smaller subgroup, and the lower triangular matrices with ones down the main diagonal form an even smaller subgroup.