Let H be the set of 2 × 2, invertible, antidiagonal matrices with realentries:H ={(0 ab 0)| a, b ∈ R, ab 6 = 0}.Does H form a group under matrix multiplication? (Justify your answer.)

To determine if H forms a group under matrix multiplication, we need to check the four group axioms:

1. Closure: If two matrices A and B are in H, then their product AB must also be in H.
2. Associativity: For any three matrices A, B, and C in H, the equation (AB)C = A(BC) must hold.
3. Identity element: There must be an identity matrix I in H such that for any matrix A in H, AI = IA = A.
4. Inverse element: For each matrix A in H, there must be an inverse matrix A^-1 in H such that AA^-1 = A^-1A = I.

Let's check each axiom:

1. Closure: The product of two antidiagonal matrices is another antidiagonal matrix. However, we need to ensure that the product of two matrices in H is also invertible and has non-zero entries on the antidiagonal. If we multiply two matrices in H, (0 a1 0) and (0 a2 0), we get (0 a1a2 0), which is in H because a1a2 is non-zero (since both a1 and a2 are non-zero). So, H is closed under multiplication.

2. Associativity: This is a known property of matrix multiplication, so it holds for matrices in H.

3. Identity element: The identity matrix under multiplication is (0 1 0), which is in H.

4. Inverse element: The inverse of a matrix (0 a 0) in H is (0 1/a 0), which is also in H because 1/a is non-zero (since a is non-zero).

Therefore, H forms a group under matrix multiplication.