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Chapter 2: Problem 24

Pivot the system about the circled element. $$ \left[\begin{array}{rrr|r} 1 & 3 & 2 & 4 \\ (2) & 4 & 8 & 6 \\ -1 & 2 & 3 & 4 \end{array}\right] $$

Short Answer

Expert verified
To pivot the system about the circled element 2 in the second row and first column, we perform the following row operations: Row1 = Row1 - (1/2) * Row2, and Row3 = Row3 + (1/2) * Row2. The resulting matrix after pivoting is: \( \left[\begin{array}{rrr|r} 1 & 1 & -2 & 1 \\ 2 & 4 & 8 & 6 \\ 0 & 4 & 9 & 7 \end{array}\right] \)

Step by step solution

01

Identify the Pivot Element

Identify the circled element in the given matrix, which will be our pivot element. In this case, the pivot element is 2, located in the second row and first column.
02

Make the Elements Above the Pivot Element Zero

In order to make the elements above the pivot element zero, we will perform the following row operation: Row1 = Row1 - (Pivot Element / 2) * Row2, which is Row1 = Row1 - (1 / 2) * Row2. This will give us a new matrix: \( \left[\begin{array}{rrr|r} 1 & 1 & -2 & 1 \\ 2 & 4 & 8 & 6 \\ -1 & 2 & 3 & 4 \end{array}\right] \)
03

Make the Elements Below the Pivot Element Zero

In order to make the elements below the pivot element zero, we will perform the following row operation: Row3 = Row3 + (Pivot Element / 2) * Row2, which is Row3 = Row3 + (1 / 2) * Row2. This will give us our final matrix after pivoting about the circled element: \( \left[\begin{array}{rrr|r} 1 & 1 & -2 & 1 \\ 2 & 4 & 8 & 6 \\ 0 & 4 & 9 & 7 \end{array}\right] \) Now, we have pivoted the system about the circled element.

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Key Concepts

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

Matrix Row Operations
Matrix row operations are a set of techniques used to manipulate matrices in various mathematical processes, including solving systems of linear equations. These operations are crucial because they allow us to simplify matrices, making them easier to work with. There are three primary row operations you can perform on a matrix:
  • Row Swapping: This involves swapping two rows of the matrix. This operation is helpful when you want to move a row with a better pivot element to a higher position.

  • Row Multiplication: You can multiply all elements of a row by a non-zero scalar. This operation is useful if you need to normalize a row, for example, making the pivot element equal to 1.

  • Row Addition: You can add or subtract multiples of one row to another row. This is the most commonly used operation for eliminating coefficients of variables in linear systems.
When performing these operations, the goal is to create zeros in certain positions of the matrix to achieve simpler forms.
Gaussian Elimination
Gaussian elimination is a systematic method used to solve linear systems by using matrix row operations to transform the system matrix into a more manageable form. The idea is to simplify a matrix towards an upper triangular form, where all elements below the main diagonal are zeros.
To achieve this, the process involves several steps:
  • Forward Elimination: This step consists of using matrix row operations to create zeros below each pivot element, forming an upper triangular matrix. This determines the main diagonal.

  • Pivot Selection: Choosing the right pivot element is vital. A good pivot element simplifies the following operations. It is typically the first non-zero element from the left in the row being manipulated.

  • Back Substitution: Once the upper triangular form is achieved, you solve for the variables starting from the last row upwards, making backward substitutions to find all unknowns.
Gaussian elimination is widely used for its robustness and simplicity in solving linear systems and finding matrix inverses.
Linear Systems Solutions
A system of linear equations is a collection of one or more linear equations involving the same set of variables. The solution to these systems involves finding values for the variables that satisfy all equations simultaneously. The matrix representation of a system of equations is an efficient way to work with these systems. Each equation corresponds to a row in the matrix, and each variable is associated with a column.
  • Consistent Systems: These systems have at least one solution. If a system can be simplified to have a pivot (non-zero leading coefficient) for each variable, it has a unique solution.

  • Inconsistent Systems: Such systems have no solutions. This occurs when the row operations result in a row of zeros equating to a non-zero number.

  • Dependent Systems: These systems have infinitely many solutions, typically involving free variables that can take any value. They occur when there are more variables than independent equations.
Understanding how to interpret these systems is key to finding practical solutions to mathematical models and real-world problems involving linear relationships.

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Most popular questions from this chapter

K & R Builders build three models of houses, \(M_{1}, M_{2}\), and \(M_{3}\), in three subdivisions I, II, and III located in three different areas of a city. The prices of the houses (in thousands of dollars) are given in matrix \(A\) : K\& R Builders has decided to raise the price of each house by \(3 \%\) next year. Write a matrix \(B\) giving the new prices of the houses.

Matrix \(A\) is an input-output matrix associated with an economy, and matrix \(D\) (units in millions of dollars) is a demand vector. In each problem,find the final outputs of each industry such that the demands of industry and the consumer sector are met. $$ A=\left[\begin{array}{lll} 0.2 & 0.4 & 0.1 \\ 0.3 & 0.2 & 0.1 \\ 0.1 & 0.2 & 0.2 \end{array}\right] \text { and } D=\left[\begin{array}{r} 6 \\ 8 \\ 10 \end{array}\right] $$

The property damage claim frequencies per 100 cars in Massachusetts in the years 2000,2001 , and 2002 are \(6.88,7.05\), and \(7.18\), respectively. The corresponding claim frequencies in the United States are 4.13, \(4.09\), and \(4.06\), respectively. Express this information using a \(2 \times 3\) matrix.

Kaitlin and her friend Emma returned to the United States from a tour of four cities: Oslo, Stockholm, Copenhagen, and Saint Petersburg. They now wish to exchange the various foreign currencies that they have accumulated for U.S. dollars. Kaitlin has 82 Norwegian krones, 68 Swedish krones, 62 Danish krones, and 1200 Russian rubles. Emma has 64 Norwegian krones, 74 Swedish krones, 44 Danish krones, and 1600 Russian rubles. Suppose the exchange rates are U.S. \(\$ 0.1651\) for one Norwegian krone, U.S. \$0.1462 for one Swedish krone, U.S. \$0.1811 for one Danish krone, and U.S. \(\$ 0.0387\) for one Russian ruble. a. Write a \(2 \times 4\) matrix \(A\) giving the values of the various foreign currencies held by Kaitlin and Emma. (Note: The answer is not unique.) b. Write a column matrix \(B\) giving the exchange rate for the various currencies. c. If both Kaitlin and Emma exchange all their foreign currencies for U.S. dollars, how many dollars will each have?

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{aligned} -x_{2}+x_{3} &=2 \\ 4 x_{1}-3 x_{2}+2 x_{3} &=16 \\ 3 x_{1}+2 x_{2}+x_{3} &=11 \end{aligned} $$
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