In the Dot Product, Cross Product, and Equations of Lines and Planes in Space sections, we will explore the applications of and how to find the dot product and cross product of two vectors. Here we will review how to evaluate the sine and cosine functions at specific angle measures and evaluate the determinant of a [latex]2\times2[/latex] matrix.

Find Reference Angles

(See Module 2, Skills Review for Vectors in the Plane)

Evaluate the Determinant of a [latex]2\times2[/latex] Matrix

A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.

A General Note: Find the Determinant of a [latex]2\times2[/latex] Matrix

The determinant of a [latex]2\text{ }\times \text{ }2[/latex] matrix, given

[latex]A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right][/latex]

is defined as

Figure 1

Notice the change in notation. There are several ways to indicate the determinant, including [latex]\mathrm{det}\left(A\right)[/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[/latex].