## Section3.9Determinants

The determinant of a (square) matrix is somewhat complicated in general, so you may want to check a reference book. The $2\times2$ and $3\times3$ cases can be memorized using the examples below.

The determinant of a $2\times2$ matrix is given by

$$\det\begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix} = \begin{vmatrix} a\amp b\\ c\amp d\\ \end{vmatrix} = ad-bc\text{.}\tag{3.9.1}$$

Notice in the equation above the two common notations for determinant.

The determinant of a $3\times3$ matrix is computed as follows:

\begin{align} \begin{vmatrix} a\amp b\amp c\\ d\amp e\amp f\\ g\amp h\amp i\end{vmatrix} \amp = \det\begin{pmatrix} a\amp b\amp c\\ d\amp e\amp f\\ g\amp h\amp i \end{pmatrix} = a\cdot \begin{vmatrix} e\amp f\\ h\amp i \end{vmatrix} - b\cdot \begin{vmatrix} d\amp f\\ g\amp i \end{vmatrix} + c\cdot \begin{vmatrix} d\amp e\\ g\amp h \end{vmatrix}\notag\\ \amp = a\cdot(ei-hf)-b\cdot(di-gf)+c\cdot(dh-ge)\notag\\ \amp = aei-ahf-bdi+bgf+cdh-cge\text{.}\tag{3.9.2} \end{align}

The smaller $2\times2$ determinants are called the cofactors of the elements $a\text{,}$ $b\text{,}$ and $c\text{,}$ respectively. The minus sign in front of $b$ is part of the cofactor. Cofactors are formed by keeping only what is left after eliminating everything from the row and column where the element desired resides. So, for $a\text{,}$ the row elements, $b$ and $c\text{,}$ and the column elements, $d$ and $g\text{,}$ are eliminated, leaving the $2\times2$ matrix shown above.

Computing $4\times4$ matrices is a straightforward extension of the above procedure, but it is easier to just go to a computer!!!

Try it for yourself by determining

$$\det\left(\begin{array}{ccc} 1\amp 2\amp 3\\ 4\amp 5\amp 6\\ 7\amp 8\amp 9 \end{array} \right)\tag{3.9.3}$$