A regular matrix problem

[mahjk17]

I am very raw at proofs, I only had experience with them one semester with very few proof related questions. The question states, " suppose A=LU is a regular matrix. Write down the LU factorization of A^T. Prove that A^T is also regular, and its pivots are the same as the pivots of A."
I know for a matrix to be regular there can't be any free variables and since the questions states that A is regular so must LU since its equal to A, but I don't know how to explain that A^T is also regular and how its pivots line up the same as A. I have done an example using numbers and it turns out to be true but how can you express it formally using a proof?

 

[daon2]

The transpose of a product is the product of the transposes in reverse order.

The transpose of a lower triangular matrix is upper triangular and vice versa.

The determinant of a square matrix A is equal to the determinant of A^T.

 

[mahjk17]

Thank you very much again daon2 !!