Question

Why must the transpose of an invertible matrix be invertible?

Answer 1

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`