Determinant on 3x3 matrix and above

Question

When finding the determinent of a matrix, what is the rationale behind multiplying the entry along the row we are deleting from times the cofactor expansion? Also how does doing cofactor expansion over and over again until we reach a 2x2 fit into the picture of ad-bc? I get that we delete a row/column to reduce the size of the matrix so that we can eventually solve a 2x2 but I'm just not seeing how it makes sense besides "it just works". I have seen how this row/column deletion works on a 2x2 matrix and it makes sense but I do not see how the technique follows through on anything bigger. How does the technique give us ad-bc?

Answer 1

You might be thinking about this from the wrong direction:

The "determinant" is a quantity that has roots in Gaussian elimination. When you perform Gaussian elimination on a matrix, a certain quantity shows up, which mathematicians took to calling the "determinant"; the determinant has a certain definition derived from the process of Gaussian elimination. It just so happens that the correct formula for a 2x2 matrix is ad-bc.

Mathematicians didn't come up with the 2x2 determinant first, and then try to extend the concept to bigger matrices.

We use the cofactor expansion idea, because that's what lines up with the "determinant" quantity from Gaussian elimination.