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Chapter 3: Problem 16

Use Cramer's Rule to solve (if possible) the system of linear equations. $$\begin{aligned} -0.4 x_{1}+0.8 x_{2} &=1.6 \\ 0.2 x_{1}+0.3 x_{2} &=0.6 \end{aligned}$$

Short Answer

Expert verified
The solution to the system is \(x_{1} = 0\), \(x_{2} = 2\).

Step by step solution

01

Compute the coefficient matrix and its determinant

The coefficient matrix \(A\) of the system is \(\begin{bmatrix} -0.4 & 0.8 \\ 0.2 & 0.3 \end{bmatrix}\). Its determinant, \(D\), can be computed as follows: \[ D = (-0.4)(0.3) - (0.2)(0.8) = -0.12 - 0.16 = -0.28 \]
02

Compute the determinants of the altered matrices

First, replace the first column of matrix \(A\) with the constants, forming matrix \(A_{1}\) which becomes \(\begin{bmatrix} 1.6 & 0.8 \\ 0.6 & 0.3 \end{bmatrix}\). Compute the determinant, \(D_{1}\) from this matrix: \[ D_{1} = (1.6)(0.3) - (0.6)(0.8) = 0.48 - 0.48 = 0. \] Second, replace the second column of matrix \(A\) with the constants, forming matrix \(A_{2}\) as \(\begin{bmatrix} -0.4 & 1.6 \\ 0.2 & 0.6 \end{bmatrix}\). Compute the determinant \(D_{2}\) from this matrix: \[ D_{2} = (-0.4)(0.6) - (0.2)(1.6) = -0.24 - 0.32 = -0.56. \]
03

Compute the solutions

The solutions \(x_{1}\) and \(x_{2}\) can be calculated by dividing the determinants \(D_{1}\) and \(D_{2}\) by \(D\). This gives: \[ x_{1} = \frac{D_{1}}{D} = \frac{0}{-0.28} = 0, \] \[ x_{2} = \frac{D_{2}}{D} = \frac{-0.56}{-0.28} = 2. \] So the solution to the system is \(x_{1} = 0\), \(x_{2} = 2\).

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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 two or more linear equations involving the same set of variables. Understanding how to solve these systems is crucial for many fields, including engineering, economics, and computer science. When attempting to find a solution to a system of linear equations, we are essentially looking for values of the variables that satisfy all equations simultaneously.

There are multiple methods to solve such systems, including graphing, substitution, elimination, and matrix-based approaches like Cramer's Rule. While graphical methods may provide a good visual understanding, they are not always practical, especially with more complex systems or those with many variables. Cramer's Rule offers a direct method for solving systems that have as many equations as unknowns, and where the determinant of the coefficient matrix is non-zero.
Determinant of a Matrix
The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix. It is denoted by 'det(A)' or '|A|', where 'A' is the matrix. The determinant has important properties and is instrumental in solving systems of linear equations, particularly in determining whether a unique solution exists.

The determinant can tell us if a system is solvable or not: if it's zero, the system may have no solution or an infinite number of solutions; if it's non-zero, the system has a unique solution. Calculating the determinant can be straightforward for a 2x2 matrix, but for larger matrices, it requires more complex operations such as expansion by minors or using more advanced techniques like LU decomposition.
Coefficient Matrix
In the context of systems of linear equations, the coefficient matrix is a square matrix containing the coefficients of the variables in the system. Each row of the matrix corresponds to an equation, and each column corresponds to a variable. The coefficient matrix is a powerful tool in analyzing systems because it encapsulates all the important information about the system's structure without involving the constant terms.

Understanding the coefficient matrix is essential when applying Cramer's Rule. This method involves replacing columns of the coefficient matrix with the column of constants from the equations and then calculating the determinants of these modified matrices. The values of the variables are then found by dividing these determinants by the determinant of the original coefficient matrix, provided that the original determinant is not zero.

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Most popular questions from this chapter

Software Publishing The table shows the estimated revenues (in billions of dollars) of software publishers in the United States from 2011 through 2013 . (Source: U.S. Census Bureau) $$\begin{array}{c|c}\hline \text {Year} & \text {Revenues, } y \\\\\hline 2011 & 156.8 \\\2012 & 161.7 \\\2013 & 177.2 \\\\\hline\end{array}$$ (a) Create a system of linear equations for the data to fit the curve\(y=a t^{2}+b t+c\) where \(t=1\) corresponds to \(2011,\) and \(y\) is the revenue. (b) Use Cramer's Rule to solve the system. (c) Use a graphing utility to plot the data and graph the polynomial function in the same viewing window. (d) Briefly describe how well the polynomial function fits the data.

Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. $$\begin{aligned}&x_{1}-x_{2}-x_{3}-x_{4}=0\\\&x_{1}+x_{2}-x_{3}-x_{4}=0\\\&x_{1}+x_{2}+x_{3}-x_{4}=0\\\&x_{1}+x_{2}+x_{3}+x_{4}=6\end{aligned}$$

Find the value(s) of \(k\) such that \(A\) is singular. $$A=\left[\begin{array}{rr}k-1 & 3 \\\2 & k-2\end{array}\right]$$

Let \(A\) be an \(n \times n\) nonzero matrix satisfying \(A^{10}=O .\) Explain why \(A\) must be singular. What properties of determinants are you using in your argument?

Guided Proof Prove Property 3 of Theorem 3.3 When \(B\) is obtained from \(A\) by multiplying a row of \(A\) by a nonzero constant \(c, \operatorname{det}(B)=c \operatorname{det}(A)\) Getting Started: To prove that the determinant of \(B\) is cqual to \(c\) times the determinant of \(A,\) you need to show that the determinant of \(B\) is equal to \(c\) times the cofactor expansion of the determinant of \(A\). (i) Begin by letting \(B\) be the matrix obtained by multiplying \(c\) times the \(i\) th row of \(A\) (ii) Find the determinant of \(B\) by expanding in this ith row. (iii) Factor out the common factor \(c .\) (iv) Show that the result is \(c\) times the determinant of \(A\)
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