Linear Algebra, 5th Edition

4.1 Determinants of Order 2

In this section, we define the determinant of a $2\times 2$ matrix and investigate its geometric significance in terms of area and orientation.

Definition.

If

$$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

is a $2\times 2$ matrix with entries from a field F, then we define the determinant of A, denoted det(A) or $\left|A\right|,$ to be the scalar $ad-bc$.

Example 1

For the matrices

$$A=\left(\begin{array}{ll}1& 2\\ 3& 4\end{array}\right)\text{and}B=\left(\begin{array}{ll}3& 2\\ 6& 4\end{array}\right)$$

in ${\text{M}}_{2\times 2}(R),$ we have

$$\det (A)=1\cdot 4-2\cdot 3=-2\text{and}\det (B)=3\cdot 4-2\cdot 6=0.$$

For the matrices A and B in Example 1, we have

$$A+B=\left(\begin{array}{ll}4& 4\\ 9& 8\end{array}\right),$$

and so

$$\det (A+B)=4\cdot 8-4\cdot 9=-4.$$

Since $\det (A+B)\ne \det (A)+\det (B),$ the function det: ${\text{M}}_{2\times 2}(R)\to R$ is not a linear transformation. Nevertheless, the determinant does possess an important linearity property, which is explained in the following theorem.

Theorem 4.1.

The function det: ...