Determinant Transpose Proof - Rhea

Statement: I am going to derive through a series of statements that transposing a matrix does NOT change its determinant.

First we will start with a 2x2 matrix as follows:

Let the 2x2 matrix A=

$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $

So, by calculating the determinant, we get det(A)=ad-cb, Simple enough, now lets take A^T (the =transpose).

A^T=

$ \begin{bmatrix} a & c \\ b & d \end{bmatrix} $

So, det(A^T)=ad-cb.

Well, for this basic example of a 2x2 matrix, it shows that det(A)=det(A^T). Simple enough...