Problem 25 - Precalculus with Limits

Question

In Exercises 25-34, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). \(\left[ \begin{array}{r} 1 & 2 & -1 \\ 3 & 7 & -10 \\ -5 & -7 & -15 \end{array} \right]\)

Step-by-Step Solution

Answer

The inverse of the matrix, if it exists, can be found using a graphing utility. However, without the actual computations and verification with an identity matrix, the exact inverse cannot be provided in this format.

Step 1: Load the matrix into a graphing utility

First load the given matrix into a graphing utility. The matrix to be entered is: \[ \left[ \begin{array}{r} 1 & 2 & -1 \ 3 & 7 & -10 \ -5 & -7 & -15\end{array} \right] \]

Step 2: Find the inverse

Use the 'Inverse' function of the graphing utility to find the inverse of the loaded matrix. If the inverse exists, the utility will return it. If the inverse does not exist, it will either return an error or an undefined value.

Step 3: Check the result

To verify if the obtained matrix is indeed the inverse, multiply the original matrix with its supposed inverse using the 'Multiply' function of the graphing utility. If the result is an identity matrix, then the obtained matrix is indeed the inverse.

Key Concepts

Graphing Utility

A graphing utility is a helpful tool that allows students and mathematicians to perform various operations on matrices and other mathematical functions easily. Some common graphing utilities include software programs or calculator applications capable of handling complex calculations. Here's how a graphing utility can assist you in matrix tasks:

• Inputting Matrices: Graphing utilities allow you to enter matrices directly into their interface, which is the first step in solving many matrix problems.
• Inverse Calculation: These tools often have built-in functions to calculate the inverse of a matrix if it exists.
• Verification: After finding an inverse, graphing utilities enable you to verify the result by performing matrix multiplication.

Using a graphing utility simplifies complex operations, saving time and reducing errors. When you're handling matrices, especially in exams or homework, such utilities are indispensable. By entering the matrix from the notebook, you can easily apply the inverse function and check results with minimal effort.

Inverse Matrix

An inverse matrix is a matrix that, when multiplied with its original, yields the identity matrix. Not all matrices have an inverse; a matrix, to have an inverse, must be square (same number of rows and columns) and have a non-zero determinant.

Steps to Find an Inverse Matrix

• Load the Matrix: Begin by loading your matrix into a graphing utility.
• Calculate the Determinant: Determine the determinant of the matrix. A determinant of zero means the matrix does not have an inverse.
• Use the Inverse Function: Use the graphing utility's inverse function to compute the inverse.
• Verify the Result: Confirm your answer by multiplying the original matrix and its inverse. The result should be an identity matrix.

Understanding the inverse of a matrix is crucial in solving systems of linear equations, among other applications in engineering and physics.

Identity Matrix

An identity matrix is a special type of square matrix where all the diagonal elements are one, and all other elements are zero. It plays a crucial role in checking the correctness of an inverse matrix.

Features of the Identity Matrix

• Square: An identity matrix is always square, such as 2x2, 3x3, etc.
• Multiplicative Identity: Just like the number 1 is the identity for multiplication in arithmetic, an identity matrix serves a similar purpose for matrix multiplication. Any matrix multiplied by an identity matrix results in the original matrix.

Role in Verifying Inverses

To check if a matrix's computed inverse is correct, multiply the original matrix by its inverse. If your calculation is correct, the product will be an identity matrix. This verification step is essential when working with matrices to ensure accuracy in calculations.