Let A be a symmetric invertible matrix, AT=A, A−1A=AA−1=I Can it be shown that A−1 is also symmetric?
I seem to remember a proof similar to this from my linear algebra class, but it has been a long time, and I can’t find it in my text book.
Answer
You can’t use the thing you want to prove in the proof itself, so the above answers are missing some steps. Here is a more complete proof. Given A is nonsingular and symmetric, show that A−1=(A−1)T:
I=IT
since AA−1=I,
AA−1=(AA−1)T
since (AB)T=BTAT,
AA−1=(A−1)TAT
since AA−1=A−1A=I, we rearrange the left side
A−1A=(A−1)TAT
since A=AT, we substitute the right side
A−1A=(A−1)TA
A−1A(A−1)=(A−1)TA(A−1)
A−1I=(A−1)TI
A−1=(A−1)T
and we are done.
Attribution
Source : Link , Question Author : gregmacfarlane , Answer Author : D.Deriso
