Answer to Question #197150 in Linear Algebra for Stefanus Weyulu

Question

 Prove that if A and B are matrices such that A is symmetric, then (BA−1 ) T (A−1BT ) −1 = In.

Answer 1

To prove: $(BA^{-1})^T(A^{-1}B^T)^{-1} =I$

As A is symmetric matrix $A^T=A$

Taking LHS-

$(BA^{-1})^T(A^{-1}B^T)^{-1}$

$=(A^{-1})^TB^T(B^T)^{-1}(A^{-1})^{-1}$

$=(A^{-1})^TA^T ~~~~~~~~~~~~~~~~~~~( As A^T=A)$

$=I$