Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrrr} 4 & 8 & -7 & 14 \\ 2 & 5 & -4 & 6 \\ 0 & 2 & 1 & -7 \\ 3 & 6 & -5 & 10 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix can be calculated by following the steps above using a graphing utility.This includes checking if the determinant is non-zero, and then calculating the inverse if it does exist. The answer would be a 4x4 matrix which is the inverse of the initial matrix.

Step 1: Calculating the determinant

First, we need to check if our matrix has an inverse. For a matrix to have an inverse, the determinant must not be zero. Compute the determinant of the given 4x4 matrix using your graphing utility. Let's represent the given matrix as A.

Step 2: Verify determinant is not zero

Verify that the computed determinant is not zero. If it is zero, that implies the matrix does not have an inverse and we do not proceed. Otherwise, proceed to the next step.

Step 3: Calculating the inverse

Using your graphing utility, input the matrix A into the matrix function. Then, find the option for 'inverse matrix' in your utility and select it. The output will be the inverse matrix of A, suppose we represent it by A^(-1).

Key Concepts

Determinant Calculation

Before finding the inverse of a matrix, it's crucial to determine if it even exists. For a 4x4 matrix, like the one given in the exercise, this involves calculating its determinant. The determinant is a special number that provides important information about a matrix.
A non-zero determinant indicates that the matrix has an inverse, making it invertible. If the determinant is zero, the matrix is singular, meaning it does not have an inverse. To calculate the determinant of a 4x4 matrix:

• Choose a row or column to expand across. Selecting the one with the most zeros simplifies calculations.
• Use cofactor expansion, which involves recursive calculations of smaller 3x3 determinants.
• Combine these using the signs determined by the position in the original matrix (positive or negative).

For simpler calculations, a graphing utility can provide the determinant quickly, ensuring we don't make manual errors.

Graphing Utility

A graphing utility is a powerful tool for mathematicians and students alike. It can handle complex calculations, like finding determinants and inverses, with ease. Using a graphing utility not only saves time but also minimizes the risk of arithmetic mistakes.
Most graphing calculators have built-in functions for matrices. To use a graphing utility:

• Navigate to the matrix function, typically found under a 'matrix' or 'math' menu.
• Input the matrix by entering its rows and columns sequentially.
• Choose the operation you need, like determinant or inverse calculation. The utility will do the heavy lifting for you.

It outputs the result promptly, helping you to verify your steps quickly. This ease of use makes it an invaluable tool for students handling large matrices.

4x4 Matrix

A 4x4 matrix is a square array of numbers arranged in 4 rows and 4 columns. In linear algebra, matrices like this are used to solve systems of equations, perform transformations, and more. Understanding their properties is essential for various applications. Key characteristics of a 4x4 matrix:

• Contains 16 elements.
• The main diagonal runs from the top left to the bottom right.
• Determinant calculation is more involved than smaller matrices, often requiring a cofactor expansion over a row or column.

Such matrices can become tedious to work with manually due to the complexity of calculations, which is why tools like graphing utilities are so handy for verifying results.

Inverse Matrix

The inverse of a matrix, when it exists, is an essential concept in linear algebra. The inverse serves a similar function to division in arithmetic. When you multiply a matrix by its inverse, the result is the identity matrix, which is the equivalent of the number 1 in regular math.
To find the inverse of a 4x4 matrix, ensure first that the determinant is non-zero. The process, especially using a graphing utility, involves simple steps:Steps to find an inverse using a graphing utility:

• Input the matrix into the utility as described earlier.
• Access the matrix functions and look for the inverse option.
• Select it to compute the inverse matrix, denoted as \( A^{-1} \).

Remember, if the determinant is zero, the inverse does not exist, and thus no further calculations are possible.