## Basic Matrix Features, simple $$2 \times 2$$ determinant

A simple $$2\times 2$$ determinant illustrates this. We have $$det(\mathbf{A})= \left| \begin{array}{cc} a_{11}& a_{12}\\ a_{21}&a_{22}\end{array} \right|= (-1)^0a_{11}a_{22}+(-1)^1a_{12}a_{21},$$ where in the last term we have interchanged the column indices $$1$$ and $$2$$. The natural ordering we have chosen is $$a_{11}a_{22}$$.