Let G be the set of 2 × 2, invertible, diagonal matrices with real entries:G ={(a 00 b)| a, b ∈ R, ab 6 = 0}.Show that G forms a group under matrix multiplication. (You may as-sume that matrix multiplication is associative.)

To show that G forms a group under matrix multiplication, we need to verify the four group axioms:

1. Closure: If we multiply any two 2x2 invertible, diagonal matrices with real entries, the result will also be a 2x2 invertible, diagonal matrix with real entries. This is because the multiplication of two diagonal matrices results in another diagonal matrix where the elements on the diagonal are the products of the corresponding elements on the diagonals of the original matrices. Since a and b are non-zero real numbers, their product will also be a non-zero real number. Therefore, the set G is closed under matrix multiplication.

2. Associativity: As given, matrix multiplication is associative. This means that for any three matrices A, B, and C in G, (AB)C = A(BC).

3. Identity element: The identity element in the group G under matrix multiplication is the 2x2 diagonal matrix with 1's on the diagonal (1 0 0 1). This is because multiplying any matrix in G by this identity matrix does not change the original matrix.

4. Inverse element: For each matrix in G, there exists an inverse matrix in G. The inverse of a 2x2 diagonal matrix (a 0 0 b) is (1/a 0 0 1/b), which is also in G because 1/a and 1/b are real numbers (since a and b are non-zero real numbers).

Therefore, G forms a group under matrix multiplication.