Problem 39 - Precalculus with Limits

Question

In Exercises 39-54, find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest. \(\left[ \begin{array}{r} 2 & -1 & 0 \\ 4 & 2 & 1 \\ 4 & 2 & 1 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Identify the matrix

The given matrix \(M\) is:\[\left[ \begin{array}{ccc} 2 & -1 & 0 \ 4 & 2 & 1 \ 4 & 2 & 1 \end{array} \right]\]

Step 2: Duplicate Rows

Notice that the second and third rows of the matrix are identical. This means that the determinant of the matrix is zero, as a basic property of determinants is that the determinant of a matrix with two identical rows is zero.

Step 3: Manual Calculation (optional)

However, for educational purposes, let's compute it manually:The formula for a 3x3 determinant is\[\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a \begin{vmatrix} e & f \ h & i \end{vmatrix} - b \begin{vmatrix} d & f \ g & i \end{vmatrix} + c \begin{vmatrix} d & e \ g & h \end{vmatrix}\]In this case, \(a=2, b=-1, c=0, d=4, e=2, f=1, g=4,h=2, i=1\).Computing the determinant:\[= 2 \begin{vmatrix} 2 & 1 \ 2 & 1 \end{vmatrix} - (-1) \begin{vmatrix} 4 & 1 \ 4 & 1 \end{vmatrix} + 0\]Which simplifies to \[= 2 (2*1 - 2*1) + 1 (4*1 - 4*1)= 0\]So the determinant is indeed 0. Note that we could have avoided this last step knowing that the rows were duplicated.

Key Concepts

Matrix Properties

Understanding the properties of a matrix can make working with them much easier. Matrices are rectangular arrays of numbers or expressions arranged in rows and columns. Several important properties help us simplify and solve matrix-related problems.

One of the most crucial properties of matrices regarding determinants is that if a matrix has two identical rows or columns, its determinant is zero. This is because the rows, or columns, being identical means that the volume of the "box" formed by the matrix is flattened to zero;

In essence, the matrix's capability to "span space" is lost. This property allows us to quickly determine certain matrix values without lengthy calculations and can be an invaluable tool for quickly identifying key characteristics of the matrix's behavior.

Cofactor Expansion

Cofactor Expansion, also known as Laplace's expansion, is a method for calculating the determinant of a matrix. It's particularly useful for matrices larger than 2x2. The cofactor expansion involves breaking down a complex matrix into simpler parts, making the computation of the determinant manageable.

To use cofactor expansion, you select a row or column of the matrix. Each element in that row or column is multiplied by the determinant of a smaller matrix, which results from removing the row and column of that element. This smaller matrix is known as the "minor."

Cofactor extit{sign} is essential here as well, calculated using \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element. The general formula for cofactor expansion is:

• For an element \(a_{ij}\), cofactor is \(C_{ij} = (-1)^{i+j} \times \text{det}(M_{ij})\)
• Sum these to complete the determinant.

Although identifying identical rows or columns is quicker, cofactor expansion provides a dependable method when that isn't immediately apparent.

3x3 Matrices

Matrices of size 3x3 are common in linear algebra, providing a practical complexity level for educational problems. They contain 3 rows and 3 columns, offering not only a visual representation of linear transformations but also allowing us to practice determinant calculation.

Calculating the determinant of a 3x3 matrix specifically can be done using cofactor expansion. With a 3x3 matrix:

• Select either a row or a column for expansion.
• Calculate the determinant of each 2x2 "minor" matrix obtained by omitting the current element's row and column.

An advantage of 3x3 matrices is their size, which is easier to manage manually compared to larger matrices, yet still rich with intricate properties and solving methods.

This helps build a foundation for understanding higher-dimensional matrices and transformations, an essential skill in various applications, including computer graphics, physics, and engineering.