Problem 62 - Precalculus with Limits

Question

In Exercises 55-62, use the matrix capabilities of a graphing utility to evaluate the determinant. \(\left| \begin{array}{r} -2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & -4 \end{array} \right|\)

Step-by-Step Solution

Answer

The determinant of the given matrix is 48.

Step 1: Identify the type of matrix

Looking at the given matrix, observe that the elements outside the main diagonal are all zero. This means it is a diagonal matrix.

Step 2: Utilize the property of diagonal matrix

The determinant of a diagonal matrix is the product of the diagonal elements. In this case the diagonal elements are: -2, 3, -1, 2, -4.

Step 3: Calculate the determinant

Multiply the diagonal elements together to get the determinant: -2 * 3 * -1 * 2 * -4 = 48. So, the determinant of the given matrix is 48.

Key Concepts

Matrix Capabilities

Matrix capabilities refer to the powerful functionalities related to performing various operations directly on matrices, often with the help of technological tools. These capabilities empower users to:

• perform basic arithmetic operations, such as addition and multiplication,
• find the inverse or determinant of a matrix, and
• solve systems of linear equations.

When using a graphing utility with matrix capabilities, you can simplify complex manual calculations.
This is especially helpful when dealing with large matrices, where computational speed and accuracy are critical.
With these tools, you can:

• input matrices directly into the calculator or software,
• use built-in functions to perform matrix operations efficiently, and
• visualize matrices for a better understanding of their structure.

Developing an understanding of matrix capabilities not only simplifies solving mathematical problems but also deepens comprehension of matrix operations.

Diagonal Matrix

A diagonal matrix is a special kind of matrix where all the elements outside the main diagonal are zero.
The main diagonal itself can be made up of zeros or any other numbers.
This structure makes diagonal matrices easier to work with in mathematical calculations. Some interesting properties of diagonal matrices include:

• The inverse of a diagonal matrix is another diagonal matrix, assuming none of the diagonal elements are zero.
• When you multiply diagonal matrices together, the product is another diagonal matrix.
• Finding the determinant of a diagonal matrix is straightforward due to a unique property: simply multiply all the diagonal elements together.

This simplification saves a lot of computation time and effort compared to working with general matrices.
Understanding how diagonal matrices function within linear algebra can make complex calculations much more approachable.

Graphing Utility

A graphing utility is a tool, usually either a calculator or a software application, that assists in performing and visualizing complex mathematical processes.
Graphing utilities often include features such as graph plotting, equation solving, and matrix operations. For matrices, especially larger ones, using a graphing utility can:

• Provide instant calculations for determinant, inverse, and other operations.
• Help visualize matrices and plot graphs for better understanding of math concepts.
• Perform what might be complicated algebraic manipulations manually, more accurately and swiftly.

This accessibility to powerful computational tools empowers students and professionals alike to solve matrix-related problems more effectively.
Using a graphing utility also encourages explorative learning by allowing users to test and verify mathematical theories and concepts on the fly.