Is the inverse of a symmetric matrix also symmetric?

Let A be a symmetric invertible matrix, AT=A, A1A=AA1=I Can it be shown that A1 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 A1=(A1)T:

I=IT

since AA1=I,

AA1=(AA1)T

since (AB)T=BTAT,

AA1=(A1)TAT

since AA1=A1A=I, we rearrange the left side

A1A=(A1)TAT

since A=AT, we substitute the right side

A1A=(A1)TA
A1A(A1)=(A1)TA(A1)
A1I=(A1)TI
A1=(A1)T

and we are done.

Attribution
Source : Link , Question Author : gregmacfarlane , Answer Author : D.Deriso

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