Skills Review for Change of Variables in Multiple Integrals

Learning Outcomes

  • Evaluate the determinant of a 2×2 Matrix
  • Evaluate the determinant of a 3×3 Matrix

In the Change of Variables in Multiple Integrals section, we will explore how to use change of variables to calculate double and triple integrals. Here we will review how to evaluate the determinant of 2 x 2 and 3 x 3 matrices.

Evaluate the Determinant of a 2×2 Matrix

(also in Module 2, Skills Review for the Dot Product, Cross Product, and Equations of Lines and Planes in Space)
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 2 × 2 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

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].

Example: Finding the Determinant of a 2 × 2 Matrix

Find the determinant of the given matrix.

[latex]A=\left[\begin{array}{cc}5& 2\\ -6& 3\end{array}\right][/latex]

Try It

Evaluate the Determinant of a 3 × 3 Matrix

Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.

Find the determinant of the 3×3 matrix.

[latex]A=\left[\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}\right][/latex]
  1. Augment [latex]A[/latex] with the first two columns.
    [latex]\mathrm{det}\left(A\right)=\left\rvert\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}\right\rvert \left.\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right\rvert[/latex]
  2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
  3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.

The algebra is as follows:

[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[/latex]

Example: Finding the Determinant of a 3 × 3 Matrix

Find the determinant of the given 3 × 3 matrix:

[latex]A=\left[\begin{array}{ccc}0& 2& 1\\ 3& -1& 1\\ 4& 0& 1\end{array}\right][/latex]

Try It

Find the determinant of the 3 × 3 matrix.

[latex]\mathrm{det}\left(A\right)=\left\rvert\begin{array}{ccc}1& -3& 7\\ 1& 1& 1\\ 1& -2& 3\end{array}\right\rvert[/latex]

Try It