Problem 68 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 65 and 66, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$ \left[\begin{array}{rrr} 1 & 0 & 2 \\ 1 & 1 & 1 \\ 2 & -1 & 0 \end{array}\right] \quad \begin{aligned} &x=5 \\ &y=-2 \\ &z=1 \end{aligned} $$

Step-by-Step Solution

Answer

To solve the system, we must first verify that the matrix is invertible, then calculate its inverse and finally multiply it by the vector of solutions. This allows us to find the values of x, y, and z.

Step 1: Identify The Matrix and the Vector

Our first step is to correctly identify the coefficient matrix and the vertex representing the solutions on the right-hand side. So, the coefficient matrix is \[ \left[\begin{array}{rrr}1 & 0 & 2 \ 1 & 1 & 1 \ 2 & -1 & 0 \end{array}\right] \] and the vector is \[\begin{aligned}&x=5 \&y=-2 \&z=1 \end{aligned}\]

Step 2: Matrix Multiplication

The system can now be written in the following representation: \[Ax = b\]Where:* \(A\) is the coefficient matrix* \(x\) is the vector of variables (x, y, z)* \(b\) would be the solutions.By multiplying the inverse of \(A\) to both sides of the equation, we can find \(x\). However, first, we need to ensure there is an inverse. The determinant is not zero, thus it's invertible.

Step 3: Calculate the Inverse of Matrix A and Find the solution

If a graphing calculator is available, use it to find the inverse of matrix A. Multiply this inverse to matrix \(b\) to get the solution:\[x = A^{-1}b\]It should be noted that each entry in vector \(x\) corresponds to the value of the variables (x, y, z). The first entry of the latter refers to x, the second to y, and the third to z.

Key Concepts

Matrix Capabilities

Matrices are powerful mathematical tools used in many areas, including system of linear equations. With a matrix, we can succinctly represent multiple linear equations at once. This is especially useful when solving systems of equations using a graphing utility. Such devices have built-in functionalities to handle matrix operations, simplifying the problem-solving process.

Matrix capabilities enable you to perform various operations, such as:

• Addition: Combining two matrices element by element.
• Subtraction: Similar to addition, but subtract the elements instead.
• Multiplication: This can either be matrix by matrix or matrix by a vector, crucial for solving linear equations.
• Transposition: Flipping a matrix over its diagonal, turning rows into columns and vice versa.
• Determinant and Inversion: Essential for understanding if a matrix can be inverted, which is used in solving equations directly.

These operations help reveal solutions that otherwise involve lengthy algebraic steps. By utilizing matrix capabilities, complex linear systems become manageable, aiding not just solutions but also visualization using graphing tools.

Graphing Utility

A graphing utility is a versatile device or software that assists in visualizing mathematical concepts. For students tackling systems of linear equations, graphing utilities prove invaluable. They can handle tasks like plotting graphs, solving equations, and performing matrix operations.

When dealing with linear equations, a graphing utility can:

• Plot Graphs: It visually represents equations to find intersections or verify solutions.
• Display Matrices: This aids in understanding the structure and manipulations of matrices.
• Perform Calculations: Include finding matrix inverses or determinants seamlessly.
• Simulate Solution Steps: Showing step-by-step processes virtually, boosting comprehension.

These features ease the process and save time, allowing students to focus on understanding concepts instead of getting bogged down in calculations. Plus, seeing graphical representations enhances their intuition about equation behaviors and solutions.

Matrix Inversion

Matrix inversion is a process that finds a matrix, referred to as the inverse, which, when multiplied with the original matrix, results in the identity matrix. The identity matrix is akin to the number 1 in basic arithmetic; it doesn't change a matrix when multiplied by it.

To understand more:

• Prerequisites for Inversion: A matrix must be square (same number of rows and columns) and have a non-zero determinant.
• Finding the Inverse: In algebra, this may involve adjoints and determinants, but graphing calculators simplify by computing it directly.
• Applications: Inversion is primarily used to solve equations written in matrix form \(Ax = b\), where \(A^{-1}b\) gives vector \(x\).

Through matrix inversion, complex problems like systems of equations turn straightforward. It allows solutions to be directly computed without tedious manual steps, which is invaluable when working quickly or with extensive data sets.

Coefficient Matrix

A coefficient matrix is an essential structure in representing systems of equations in matrix form. It consists only of the coefficients of variables from each equation, arranged systematically. This matrix provides a visual and structured method to handle multiple linear equations simultaneously.

In the given system, the coefficient matrix is represented as:\[\begin{bmatrix}1 & 0 & 2 \1 & 1 & 1 \2 & -1 & 0\end{bmatrix}\]

• Organization: Each row corresponds to an equation, and each column to a variable.
• Matrix Representation: It simplifies the presentation and manipulation of equations, showing only what's necessary for calculations.
• Use in Calculations: Acts as matrix \(A\) in the equation \(Ax = b\) setup, essential for matrix methods in solving equations.

Understanding and using a coefficient matrix helps set a firm foundation for solving and analyzing systems of linear equations. It not only aids in defining the problem but plays a central role in applying matrix concepts and solutions efficiently.