From $\det(A^T)=\det(A)$ to equivalence of row/column expansion of determinant?

Question

(Say $A \in M_{n\times n}(F)$.) I just finished the proof of $\det(A^T)=\det(A),$ and I have two questions about this

• How can I use this fact to prove that the definition of determinant(first row expansion) is equivalent to column expansion?
• Can I mix both elementary column operation(s) and elementary row operation(s) when simplify matrix?

Appreciate any hint.

Answer 1

For your first point, notice that every operation on the rows of $A$ has an exact equivalent operation on the columns of $A^T$. This can be proved using shear matrices.

As to your second point, you can indeed mix row and column operations which can prove useful in some situations:

$$D=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 1 \\ 0 & 5 & 6 & 7 \\ 0 & 8 & 9 & 10 \end{bmatrix} $$

This examples naturally leads you to expand on both rows and columns.