Consider the set H of all 3 × 3 matrices of the type where a, b, c, d, e and f…

To determine if the set H forms a group under matrix multiplication, we check the four group axioms: closure, associativity, identity, and inverses. Matrix multiplication is associative, so that holds. The identity matrix is in H (set a=b=c=1), satisfying the identity axiom. For closure, multiplying two matrices in H results in another matrix of the same form with non-zero diagonal entries since abc ≠ 0 and the product of non-zero numbers is non-zero. For inverses, a matrix in H has determinant abc (since it's upper triangular), and since abc ≠ 0, the matrix is invertible. The inverse of an upper triangular matrix with non-zero diagonal entries is also upper triangular and has non-zero diagonal entries, so the inverse lies in H. Thus, all axioms are satisfied, and H is a group.