Problem 24 - Precalculus with Limits

Question

In Exercises 21-24, use the matrix capabilities of a graphing utility to find the determinant of the matrix. \(\left[ \begin{array}{r} 0.1 & 0.1 & -4.3 \\ 7.5 & 6.2 & 0.7 \\ 0.3 & 0.6 & -1.2 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the matrix is \(-16.75.\)

Step 1: Setting Up the Matrix

Start with the given matrix \[ \left[ \begin{array}{r} 0.1 & 0.1 & -4.3 \ 7.5 & 6.2 & 0.7 \ 0.3 & 0.6 & -1.2 \end{array} \right] \]

Step 2: Calculate Minor of the First element of the matrix

The minor of the first element (0.1) is the determinant of the 2x2 matrix that remains after the row and column (first in both cases) containing 0.1 are deleted. This results in the matrix: \[ \left[ \begin{array}{r} 6.2 & 0.7 \ 0.6 & -1.2 \end{array} \right] \]. Calculate the determinant as (6.2 * -1.2) - (0.7 * 0.6).

Step 3: Compute the Determinant

Continue with this process for other elements. The formula for calculating determinant of a 3x3 matrix is given by: Det \(A = A_{11}(A_{22}A_{33} − A_{32}A_{23}) − A_{12}(A_{21}A_{33} − A_{31}A_{23}) + A_{13}(A_{21}A_{32} − A_{31}A_{22})\). Apply this formula for our matrix.

Key Concepts

Understanding a 3x3 Matrix

A 3x3 matrix is an essential mathematical structure in the field of linear algebra. It consists of three rows and three columns, patterning nine elements in total. Notably, each element of the matrix is specified by two subscript numbers, for instance, the element in the second row and third column is designated as \(A_{23}\).

In a 3x3 matrix such as \[\left[ \begin{array}{ccc}0.1 & 0.1 & -4.3 \7.5 & 6.2 & 0.7 \0.3 & 0.6 & -1.2 \\end{array} \right]\]the structure facilitates operations such as addition, subtraction, and notably, the computation of determinants. The determinant is particularly useful and reflects properties like invertibility.

When encountering problems requiring the calculation of a matrix determinant, it's beneficial to understand how each minor contributes to the overall determination. Grasping this leads to easier computation and application in solving systems of linear equations or transformations in geometric analysis.

Exploring Matrix Minors

A matrix minor is a smaller part of a matrix formed by deleting one row and one column. For a 3x3 matrix, such as the one we discussed, the minor of an element is the determinant of the 2x2 matrix that results after removing the row and column that contains that element.

When calculating the determinant for a 3x3 matrix, you need to compute these minors first. For instance, to find the minor of the first element in the given matrix (0.1), delete the first row and first column to form the 2x2 matrix:\[\left[ \begin{array}{cc}6.2 & 0.7 \0.6 & -1.2 \\end{array} \right]\]Then, calculate the minor as the determinant of this matrix, using the formula:

• Multiply the cross-products: \((6.2 \times -1.2) - (0.7 \times 0.6)\)
• Simplify these calculations to understand how each minor affects the overall determinant.

This approach is repeated for other elements of the matrix to fully determine the entire 3x3 determinant, making it a step-by-step process involving each minor.

Using Graphing Utility for Determinants

Graphing utilities, like graphing calculators or software, are powerful tools for visualizing and computing complex mathematical problems, including matrix operations. When tasked with finding determinants of larger matrices, these utilities can save time and reduce the risk of error.

To use a graphing utility effectively to find a determinant as in the given exercise:

• Input the matrix into the utility. This typically involves selecting the matrix menu on the device or software and entering each number into the 3x3 grid provided.
• Select the option to calculate the determinant. This may involve navigating to a menu labeled "Matrix Operations" or "Determinants".
• Let the utility compute, which applies the determinant formula without manual calculation, providing the result instantly.

Using graphing utilities for matrix calculations often simplifies the learning process, allowing students to focus on understanding the results rather than manually computing each step, making it an excellent educational aid.