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Chapter 9: Problem 71

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{aligned} x-y+z &=-6 \\ 4 x+2 y+z &=9 \\ 4 x-2 y+z &=-3 \end{aligned}\right. $$

Short Answer

Expert verified
The solution for the system of equations is \(x = 1, y = 2, and z = -3\).

Step by step solution

01

Formulate Matrix A and B

The first step is to rewrite the system of equations in the matrix form \(AX = B\). For Matrix A, the coefficients of the variables x, y and z in each equation form the rows of the matrix. And Matrix B is formed by the constant term on the right side of each equation. Here, Matrix A and B will be:\(A=\left[\begin{array}{lll}1 & -1 & 1 \ 4 & 2 & 1 \ 4 & -2 & 1\end{array}\right]\), \(B=\left[\begin{array}{c}-6 \ 9 \ -3\end{array}\right]\)
02

Calculate the inverse of Matrix A

Matrix inverse is such that when it is multiplied by original matrix, it results in the identity matrix. We can use calculating function in the graphic utility to get \(A^{-1} = \left[\begin{array}{lll}0 & 0.5 & 0.5 \ -0.333 & 0.167 & -0.167 \ 0.333 & 0.833 & -0.833\end{array}\right]\)
03

Find the Solution of the System

The solution \(X\) of the given system is obtained by multiplying the inverse of Matrix A with Matrix B. Hence, \(X = A^{-1}B\). On computing we get \(X = \left[\begin{array}{c}1 \ 2 \ -3\end{array}\right]\)

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Key Concepts

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

Systems of Linear Equations
A system of linear equations consists of multiple equations with multiple variables that are interdependent. Each equation represents a line, and the solution is the point at which these lines intersect.
For the given system:
  • Equation 1: \( x - y + z = -6 \)
  • Equation 2: \( 4x + 2y + z = 9 \)
  • Equation 3: \( 4x - 2y + z = -3 \)

The goal is to find the values of \(x\), \(y\), and \(z\) that satisfy all equations simultaneously. In matrix algebra, these equations are combined into matrices which help visualize and solve the system more systematically.
Matrix Inversion
Matrix inversion is a crucial technique in solving systems of linear equations. Given a square matrix \(A\), its inverse \(A^{-1}\) is a matrix such that when multiplied with \(A\), the result is the identity matrix, denoted by \(I\).
For example, considering the matrix \(A\):\[A=\left[\begin{array}{ccc}1 & -1 & 1 \4 & 2 & 1 \4 & -2 & 1\end{array}\right]\]
Its inverse \(A^{-1}\) is calculated:\[A^{-1} = \left[\begin{array}{ccc}0 & 0.5 & 0.5 \-0.333 & 0.167 & -0.167 \0.333 & 0.833 & -0.833\end{array}\right]\]
This inversion process often involves advanced computational techniques and is facilitated using graphing utilities or algebraic software due to its complexity.
Graphing Utility Solutions
Using a graphing utility can greatly simplify the process of solving complex systems of equations and calculating matrix inverses.
A graphing utility allows input of matrices directly and utilizes its built-in functions to compute necessary operations like matrix inversion. For our system, once inputs are made as described before:
  • Matrix \(A\) is input with its coefficients
  • Matrix \(B\) is input with its constants
  • The utility computes \(A^{-1}\) and then multiplies it with \(B\)

This results in the solution matrix \(X\), where each entry corresponds to one of the variables in the system. In this case, matrix \(X\) was found to be:\[X = \left[\begin{array}{c}1 \2 \-3\end{array}\right]\]Thus, \(x = 1\), \(y = 2\), and \(z = -3\) are the solutions to the system.

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Most popular questions from this chapter

Find the quadratic function \(f(x)=a x^{2}+b x+c\) for which \(f(-2)=-4, f(1)=2,\) and \(f(2)=0\)

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Imagine the entire global population as a village of precisely 200 people. The bar graph shows some numeric observations based on this scenario. (graph can't copy) Combined, there are 183 Asians, Africans, Europeans, and Americans in the village. The number of Asians exceeds the number of Africans and Europeans by \(70 .\) The difference between the number of Europeans and Americans is \(15 .\) If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by \(23 .\) Determine the number of Asians, Africans, Europeans, and Americans in the global village.

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Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ A(B+C) $$

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{r} {2} \\ {-4} \\ {1} \end{array}\right], \quad B=\left[\begin{array}{r} {-5} \\ {3} \\ {-1} \end{array}\right] $$
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