Why must the transpose of an invertible matrix be invertible?

1 Answer
George C.
Nov 7, 2015

If A has inverse A^(-1) then A^T has inverse (A^(-1))^T

Explanation:

If you are happy to accept that A^TB^T = (BA)^T and I^T = I, then the proof is not difficult:

Suppose A is invertible with inverse A^(-1)

Then:

(A^(-1))^T A^T = (A A^(-1))^T = I^T = I

A^T (A^(-1))^T = (A^(-1) A)^T = I^T = I

So (A^(-1))^T satisfies the definition for being an inverse of A^T

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