Transpose of inverse vs inverse of transpose

I can’t seem to find the answer to this using Google.

Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix?

Answer

Is (A1)T=(AT)1 you ask.

Well
AT(A1)T=(A1A)T=IT=I(A1)TAT=(AA1)T=IT=I

This proves that the inverse of AT is (A1)T. So the answer to your question is yes.

Here I have used that
ATBT=(BA)T.
And we have used that the inverse of a matrix A is exactly (by definition) the matrix B such that AB=BA=I.

Attribution
Source : Link , Question Author : Void Star , Answer Author : Thomas

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