If is a non-singular matrix such that and , then is equal to
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: B
Solution
B = A⁻¹Aᵀ, so Bᵀ = (A⁻¹Aᵀ)ᵀ = A(Aᵀ)⁻¹ = A(A⁻¹)ᵀ.
BBᵀ = (A⁻¹Aᵀ)(A(A⁻¹)ᵀ) = A⁻¹(AᵀA)(A⁻¹)ᵀ.
Since AAᵀ = AᵀA, substitute: = A⁻¹(AAᵀ)(A⁻¹)ᵀ = (A⁻¹A)(Aᵀ(A⁻¹)ᵀ) = I·(Aᵀ(Aᵀ)⁻ᵀ)... more cleanly: A⁻¹(AAᵀ)(Aᵀ)⁻¹·...
Let me redo cleanly: BBᵀ = A⁻¹(AᵀA)(A⁻¹)ᵀ = A⁻¹(AAᵀ)(Aᵀ)⁻¹ = (A⁻¹A)(Aᵀ(Aᵀ)⁻¹) = I·I = I.
BBᵀ = (A⁻¹Aᵀ)(A(A⁻¹)ᵀ) = A⁻¹(AᵀA)(A⁻¹)ᵀ.
Since AAᵀ = AᵀA, substitute: = A⁻¹(AAᵀ)(A⁻¹)ᵀ = (A⁻¹A)(Aᵀ(A⁻¹)ᵀ) = I·(Aᵀ(Aᵀ)⁻ᵀ)... more cleanly: A⁻¹(AAᵀ)(Aᵀ)⁻¹·...
Let me redo cleanly: BBᵀ = A⁻¹(AᵀA)(A⁻¹)ᵀ = A⁻¹(AAᵀ)(Aᵀ)⁻¹ = (A⁻¹A)(Aᵀ(Aᵀ)⁻¹) = I·I = I.
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