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If a is invertible symmetric matrix, then a-1 is equal to

Answer»

Let A be a symmetric INVERTIBLE matrix, AT=A, A−1A=AA−1=I Can it be shown that A−1 is also symmetric?



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)TAA−1A(A−1)=(A−1)TA(A−1)A−1I=(A−1)TIA−1=(A−1)T

and we are done.


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