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

Haven 2020-12-30 Answered
If A,B are symmetric matrices, then prove
that (BA1)T(A1BT)1=I
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Expert Answer

Luvottoq
Answered 2020-12-31 Author has 95 answers

Step 1
Given that, A,B are symmetric matrices.
Prove that (BA1)T(A1BT)1=I
Step 2
Consider the LHS,
(BA1)T(A1BT)1=(A1)T(B)T(BT)1(A1)1
=(A1)TIA    [(BT)(BT)1=I and (A1)1=A]
=(A1)TAT    [A=AT]
=(AT)1AT    [(AT)1=(A1)T]
=I    [(AT)(AT)1=I]
=RHS
Hence , the required is obtained.

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Jeffrey Jordon
Answered 2022-01-22 Author has 2047 answers

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