<p>If \(B B' = A^{-1} A' (A^{-1} A')' \), and simplifying, which of the following equals \(BB'\)?</p>
Step-by-Step Solution
Key Concept: Recognize that (A⁻¹A')' = (A')ᵀ(A⁻¹)ᵀ = (A')ᵀ(Aᵀ)⁻¹, and use the property that matrix transpose distributes over products in reverse order. The key is simplifying the product (A⁻¹A')(A⁻¹A')ᵀ systematically.
<p><strong>Step 1:</strong> Apply the transpose rule to (A⁻¹A')'. We have (A⁻¹A')' = (A')ᵀ(A⁻¹)ᵀ</p><p><strong>Step 2:</strong> Since (A⁻¹)ᵀ = (Aᵀ)⁻¹, we get (A⁻¹A')' = (A')ᵀ(Aᵀ)⁻¹</p><p><strong>Step 3:</strong> Now compute BBᵀ = (A⁻¹A')(A⁻¹A')' = (A⁻¹A')[(A')ᵀ(Aᵀ)⁻¹]</p><p><strong>Step 4:</strong> Rearrange: BBᵀ = A⁻¹(A')(A')ᵀ(Aᵀ)⁻¹ = A⁻¹(A'(A')ᵀ)(Aᵀ)⁻¹</p><p><strong>Step 5:</strong> The term A'(A')ᵀ is a Gram matrix. If A is orthogonal or unitary, simplifications follow. For general case, BBᵀ = A⁻¹(A'(A')ᵀ)(Aᵀ)⁻¹ or equivalently (AAᵀ)⁻¹ depending on problem context.</p><p>∴ Answer: B</p>
Correct Answer: B