Differential Equations and Linear Algebra, 4th Edition

3.6 Determinants

In Theorem 2 of Section 3.5 we saw that the $2\times 2$ matrix

$$\mathbf{\text{A}}=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$$

is invertible if and only if $ad-bc\ne 0.$ The number $ad-bc$ is called the determinant of the $2\times 2$ matrix A. There are several common notations for determinants:

$$\det \mathbf{\text{A}}=\det \left[\begin{array}{ll}a& b\\ c& d\end{array}\right]=\left|\begin{array}{ll}a& b\\ c& d\end{array}\right|=ad-bc.$$ (1)

In particular, note the vertical bars that distinguish a determinant from a matrix. Sometimes we write $\det (\mathbf{\text{A}})$ to emphasize that (1) defines a function which associates a number $\left|\mathbf{\text{A}}\right|$ with each $2\times 2$ matrix A.

Example 1

$$\left|\begin{array}{rr}3& 7\\ 4& -6\end{array}\right|=(3)(-6)-(4)(7)=-46$$

Determinants are often used in elementary algebra to solve a $2\times 2$ system of the form

$$\begin{array}{rcl}{a}_{11}x+a_{12}y& =& b_1\\ a_{21}x+a_{22}y& =& b_2.\end{array}$$ (2)

It follows from Theorems 2 and 7 in Section 3.5 that this system has a unique solution if and only if its coefficient determinant ...