Question

If A,B are symmetric matrices, then prove
that \((BA^{-1})^T(A^{-1}B^T)^{-1} = I\)

Answer 1

Step 1
Given that, A,B are symmetric matrices.
Prove that \((BA^{-1})^T(A^{-1}B^T)^{-1} = I\)
Step 2
Consider the LHS,
\((BA^{-1})^T(A^{-1}B^T)^{-1} = (A^{-1})^T(B)^T(B^T)^{-1}(A^{-1})^{-1}\)
\(=(A^{-1})^T IA \ \ \ \ \left[ \because (B^T)(B^T)^{-1}=I \text{ and } (A^{-1})^{-1}=A \right]\)
\(=(A^{-1})^T A^T \ \ \ \ \left[ \because A=A^T \right]\)
\(=(A^{T})^{-1} A^T \ \ \ \ \left[ \because (A^T)^{-1}=(A^{-1})^T \right]\)
\(=I \ \ \ \ \left[ \because (A^T)(A^T)^{-1}=I \right]
\(=RHS\)
Hence , the required is obtained.