The determinant of a 3 x 3 matrix

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

is given by

$$det(A)  = a_{11}\left| \begin{matrix} a_{22} \ a_{23} \\ a_{32} \ a_{33}\end{matrix}\right| - a_{12}\left| \begin{matrix} a_{21} \ a_{23} \\ a_{31} \ a_{33}\end{matrix}\right| + a_{13}\left| \begin{matrix} a_{21} \ a_{22} \\ a_{31} \ a_{32}\end{matrix}\right|$$

This is one way of finding the determinant of a matrix. We can choose any row or column, provided it corresponds with the sign matrix, to calculate the determinant.

$$\text{Sign matrix} = \begin{pmatrix}+ \ - \ + \\ -\ + \ - \\ + \ - \ + \end{pmatrix}$$

For further information see Section 4 of the Chapter 10 Notes.