Matrix Algebra, Determinant for higher dimensions. Why -b?

I’m not sure why there is a negative infront of the “b” so I wouldn’t know how to translate this to even higher dimension matrices. An excerpt of the explanation I’m looking at is below.

First, we select a row or column. For the first value in that row, we “hide” the other values in that row and in that column, select the rest of the elements as the minor matrix, and multiply the scalar value with the determinant of the minor matrix. We repeat this for the remaining values in the first row. This diagram helps illustrate this much clearer:

3D Matrix Determinant One

The proof for this is a lengthy one and requires some more knowledge related to determinants and linear algebra that is not really required, I think, and that’s why they haven’t gone into explaining it.

For higher dimensions (n > 3) it gets more complicated but that pattern is the same.

For 3x3, it will be + - +, for 4x4 it will be + - + - and so on.

We end up relying on computational approaches instead.

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Thanks for your response, mate :slight_smile: