17 Chapter 2: Solving linear systems by matrix theory

Determinants To obtain the determinant of

A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{bmatrix}

recall the chess-board of signs

\begin{bmatrix}+&-&+\\ -&+&-\\ +&-&+\end{bmatrix},

pick one column, say the second, then multiply out with sign from above

\det A=-a_{12}\begin{vmatrix}a_{21}&a_{23}\\ a_{31}&a_{33}\end{vmatrix}+a_{22}\begin{vmatrix}a_{11}&a_{13}\\ a_{31}&a_{33}\end{vmatrix}-a_{32}\begin{vmatrix}a_{11}&a_{13}\\ a_{21}&a_{23}\end{vmatrix}

where the $2\times 2$ determinants exclude the row and column of their coefficient.