Question

A and B are non singular n x n matrices.
Prove (AᵗBᵗ)⁻¹ = (A⁻¹B⁻¹)ᵗ

Answer 1

To prove that (AᵗBᵗ)⁻¹ = (A⁻¹B⁻¹)ᵗ for non-singular n x n matrices A and B, we use two properties: the transpose of a product of matrices is the product of their transposes in reverse order, and the inverse of a product is the product of the inverses in reverse order. Applying these, we find that the statement is indeed true.

Step-by-step explanation:

To prove that (AᵄBᵄ)⁻¹ = (A⁻¹B⁻¹)ᵄ for non-singular n x n matrices A and B, we need to use the property that the inverse of a product of matrices is the product of their inverses in reverse order and the property that the transpose of a product of matrices is the product of their transposes in reverse order.

Following these properties, we have:

1. The transpose of a product is the product of transposes in reverse order: (AB)ᵄ = BᵄAᵄ.
2. The inverse of a product is the product of the inverses in reverse order: (AB)⁻¹ = B⁻¹A⁻¹.
3. Applying these properties in our case: (AᵄBᵄ)⁻¹ = (Bᵄ)⁻¹(Aᵄ)⁻¹ = B⁻¹A⁻¹ since A and B are non-singular.
4. Now taking the transpose of both sides: ((B⁻¹A⁻¹)ᵄ = (A⁻¹)ᵄ(B⁻¹)ᵄ = AᵄBᵄ, which shows the original statement to be true.

Thus, we have shown that (AᵄBᵄ)⁻¹ = (A⁻¹B⁻¹)ᵄ for non-singular n x n matrices A and B.