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Chapter 8: Problem 30

Finding the Inverse of a Matrix In Exercises \(25-34\) , use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[ \begin{array}{rrr}{-\frac{5}{6}} & {\frac{1}{3}} & {\frac{11}{6}} \\\ {0} & {\frac{2}{3}} & {2} \\ {1} & {-\frac{1}{2}} & {-\frac{5}{2}}\end{array}\right]$$

Short Answer

Expert verified
Using the graphing utility, the inverse of the provided matrix is calculated. The result depends on the use and precision of the graphing utility.

Step by step solution

01

Identify the Matrix

The given matrix is\[ \left[ \begin{array}{rrr}{-\frac{5}{6}} & {\frac{1}{3}} & {\frac{11}{6}} \ {0} & {\frac{2}{3}} & {2} \ {1} & {-\frac{1}{2}} & {-\frac{5}{2}} \end{array}\right]\]
02

Input the Matrix into the Graphing Utility

In the graphing utility, select the option to input a matrix. Input the given 3x3 matrix exactly as provided. Remember to separate each entry with a comma or space, as per the instruction of the graphing utility.
03

Calculate the Inverse

Next, select the option in your graphing utility to calculate the inverse of the matrix. If the inverse exists, the utility will display a new 3x3 matrix as the result.
04

Verify the Result

Verify the result by multiplying the given matrix with its resultant matrix (using the matrix multiplication operation in the graphing utility). If the inverse was calculated correctly, the result should be an identity matrix.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
A graphing utility is a versatile tool that goes beyond the basic calculator, allowing users to perform complex mathematical operations, including matrix calculations. When finding the inverse of a matrix, a graphing utility can greatly simplify the process:
  • First, input the matrix as provided, ensuring each element is correctly placed in the 3x3 structure.
  • The graphing utility can then be used to calculate the inverse by selecting the appropriate function. This feature automates what can otherwise be a lengthy manual calculation.
  • The inverse matrix, if it exists, will be displayed, showing you the result in a clear and concise manner.
This straightforward method greatly reduces the complexity of manually solving for an inverse, making it accessible even to those new to matrix algebra. Additionally, graphing utilities often include features to verify results, such as performing operations like matrix multiplication, which can save time and prevent errors.
Matrix Multiplication
Matrix multiplication is a fundamental operation in matrix algebra. It is especially useful for verifying solutions when finding an inverse matrix:
  • When two matrices are multiplied, the resulting matrix is determined by multiplying the rows of the first matrix by the columns of the second matrix and summing the products.
  • The process requires careful attention to ensure each element of the matrices is used correctly, often necessitating an organized approach to avoid mistakes.
  • In the context of an inverse matrix, multiplying the original matrix by its inverse should yield an identity matrix if calculated correctly. This verification step is crucial as it confirms the accuracy of the solution.
Understanding and practicing matrix multiplication is key to mastering not only inverses but many other operations within linear algebra. Utilizing a graphing utility to perform these multiplications can help confirm your manual calculations are correct.
Identity Matrix
The identity matrix plays a crucial role in matrix algebra, acting as the equivalent of the number '1' in regular arithmetic. For 3x3 matrices, the identity matrix appears as follows:
\[ \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]
  • An identity matrix is characterized by having 1s along its main diagonal, with all other elements being 0.
  • When a matrix is multiplied by its identity matrix, it remains unchanged, which is analogous to multiplying a number by 1 in basic arithmetic.
  • Confirming an inverse's correctness involves multiplying it by the original matrix to achieve this identity matrix result, providing a clear evidence of the computation's success.
Grasping the concept of an identity matrix is foundational for topics involving inverses, transformations, and other matrix-related functions. Its simplicity belies its importance in verifying that complex operations yield the correct, expected results.

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Most popular questions from this chapter

Writing (a) State Cramer's Rule for solving a system of linear equations. (b) At this point in the text, you have learned several methods for solving systems of linear equations. Briefly describe which method(s) you find easiest to use and which method(s) you find most difficult to use.

Mathematical Modeling A video of the path of a ball thrown by a baseball player was analyzed with a grid covering the TV screen. The tape was paused three times, and the position of the ball was measured each time. The coordinates obtained are shown in the table. \((x\) and \(y\) are measured in feet.) $$\begin{array}{|c|c|c|c|c|}\hline \text { Horizontal Distance, } x & {0} & {15} & {30} \\ \hline \text { Height, y } & {5.0} & {9.6} & {12.4} \\\ \hline\end{array}$$ $$\begin{array}{l}{\text { (a) Use a system of equations to find the equation of the }} \\ {\text { parabola } y=a x^{2}+b x+c \text { that passes through the }} \\ {\text { three points. Solve the system using matrices. }} \\\ {\text { (b) Use a graphing utility to graph the parabola. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Graphically approximate the maximum height of the }} \\ {\text { ball and the point at which the ball struck the ground. }} \\\ {\text { (d) Analytically find the maximum height of the ball }} \\\ {\text { and the point at which the ball struck the ground. }} \\ {\text { (e) Compare your results from parts (c) and (d). }}\end{array}$$

Breeding Facility A city zoo borrowed \(\$ 2,000,000\) at simple annual interest to construct a breeding facility. Some of the money was borrowed at \(8 \%,\) some at \(9 \%,\) and some at 12\(\% .\) Use a system of linear equations to determine how much was borrowed at each rate given that the total annual interest was \(\$ 186,000\) and the amount borrowed at 8\(\%\) was twice the amount borrowed at 12\(\% .\) Solve the system of linear equations using matrices.

Comparing Linear Systems and Matrix Operations In Exercises 41 and \(42,\) (a) perform the row operations to solve the augmented matrix, (b) write and solve the system of linear equations represented by the augmented matrix, and (c) compare the two solution methods. Which do you prefer? $$\left[ \begin{array}{rrrr}{-3} & {4} & {\vdots} & {22} \\ {6} & {-4} & {\vdots} & {-28}\end{array}\right]$$ $$\begin{array}{l}{\text { (i) Add } R_{2} \text { to } R_{1} \text { . }} \\\ {\text { (ii) Add }-2 \text { times } R_{1} \text { to } R_{2} \text { . }} \\\ {\text { (iii) Multiply } R_{2} \text { by }-\frac{1}{4}} \\ {\text { (iv) Multiply } R_{1} \text { by } \frac{1}{3}}\end{array}$$

Solving an Equation In Exercises \(81-88,\) solve for \(x .\) $$\left| \begin{array}{rr}{x+3} & {2} \\ {1} & {x+2}\end{array}\right|=0$$
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