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Chapter 9: Problem 6

Write the augmented matrix for each system and give its dimension. Do not solve. $$\begin{aligned}&3 x+5 y=-13\\\&2 x+3 y=-9\end{aligned}$$

Short Answer

Expert verified
The augmented matrix is \[\begin{pmatrix} 3 & 5 & | & -13 \ 2 & 3 & | & -9 \end{pmatrix}\] with dimensions 2x3.

Step by step solution

01

Identify Coefficients and Constants

Extract the coefficients of the variables and the constants from each equation in the system. From the first equation, the coefficients are 3 and 5, with the constant -13. From the second equation, the coefficients are 2 and 3, with the constant -9.
02

Construct the Augmented Matrix

Write the coefficients and constants in augmented matrix form. The matrix should be written as: \[\begin{pmatrix} 3 & 5 & | & -13 \ 2 & 3 & | & -9 \end{pmatrix}\].
03

Determine the Dimensions of the Matrix

Count the number of rows and columns in the matrix, excluding the line that separates the constants. The matrix has 2 rows and 3 columns. Therefore, the dimensions are 2x3.

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Key Concepts

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

system of linear equations
In mathematics, a system of linear equations is a set of equations where each equation is linear. This means each term is either a constant or the product of a constant and a single variable. Linear equations can be expressed in the following forms:
  • x + y = z
  • ax + by = c
  • 3x - 2y = 5
In our exercise, the given system is:
  • 3x + 5y = -13
  • 2x + 3y = -9
These types of equations can be represented using matrices, specifically aiming to simplify and solve them simultaneously. By converting a system of linear equations into an augmented matrix, we can use matrix operations to find solutions more easily.
matrix dimensions
Matrix dimensions refer to the size of a matrix expressed in terms of the number of rows and columns it contains. If we consider a matrix denoted as \(A\), then the dimensions would be written as \( m \times n \), where \(m \) is the number of rows and \(n \) is the number of columns. In our example problem, the augmented matrix constructed is: \begin{pmatrix} 3 & 5 & | & -13 2 & 3 & | & -9 \begin{pmatrix} To find its dimensions, we count the rows and columns. There are 2 rows and 3 columns, resulting in the dimensions of 2 \times 3. Each row represents an equation from the system, and each column represents either a coefficient or a constant from those equations.
coefficients and constants
Coefficients and constants are essential in the composition of linear equations. Coefficients are the numerical factors that multiply the variables in an equation, while constants are the standalone numbers. For instance, in the equation \( 3x + 5y = -13 \),
  • 3 and 5 are the coefficients of \(x \) and \(y \), respectively.
  • -13 is the constant.
In our given exercise, we take the coefficients and constants from each equation and arrange them into the augmented matrix. This simplifies the representation and aids in solving the system systematically. The matrix for our problem is \begin{pmatrix} 3 & 5 & | & -13 2 & 3 & | & -9 \begin{pmatrix}. Understanding how to extract these numbers from the given equations is crucial for correctly writing the augmented matrix and further matrix operations.

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Most popular questions from this chapter

Solve each problem. Tire Sales The number of automobile tire sales is dependent on several variables. In one study the relationship among annual tire sales \(S\) (in thousands of dollars), automobile registrations \(R\) (in millions), and personal disposable income \(I\) (in millions of dollars) was investigated. The results for three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b R+c l $$ where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Jarrett, J., Business Forecasting Methods, Basil Blackwell, Ltd.) (a) Substitute the values for \(S, R,\) and \(I\) for each year from the table into the equation \(S=a+b R+c I,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. (d) If \(R=117.6\) and \(I=310.73,\) predict \(S .\) (The actual value for \(S\) was \(11,314 .\) ) (e) If \(R=143.8 \text { and } I=829.06, \text { predict } S . \text { (The actual value for } S \text { was } 18,481 .)\) $$\begin{array}{|c|c|c|} \hline S & R & I \\ \hline 10,170 & 112.9 & 307.5 \\\ \hline 15,305 & 132.9 & 621.63 \\ \hline 21,289 & 155.2 & 1937.13 \\\ \hline \end{array}$$

Solve each problem. A glue company needs to make some glue that it can sell for \(120\) per barrel. It wants to use 150 barrels of glue worth \(100\) per barrel, along with some glue worth \(150\) per barrel and some glue worth \(190\) per barrel. It must use the same number of barrels of \(150\) and \(190\) glue. How much of the \(150\) and \(190\) glue will be needed? How many barrels of \(120\) glue will be produced?

Solve each problem. Several years ago, mathematical ecologists created a model to analyze population dynamics of the endangered northern spotted owl in the Pacific Northwest. The ecologists divided the female owl population into three categories: juvenile (up to \(1 \text { yr old }),\) subadult \((1\) to 2 yr old ) and adult (over 2 yr old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. \(j_{n+1}=0.33 a_{n}\) Each year 33 juvenile females are born for each 100 adult females. \(s_{n+1}=0.18 j_{n}\) Each year 18\% of the juvenile females survive to become subadults. \(a_{n+1}=0.71 s_{n}+0.94 a_{n} \quad\) Each year \(71 \%\) of the subadults survive to become adults, and \(94 \%\) of the adults survive. (a) Suppose there are currently 3000 female northern spotted owls made up of 690 juveniles, 210 subadults, and 2100 adults. Use the matrix equation on the preceding page to determine the total number of female owls for each of the next 5 yr. (b) Using advanced techniques from linear algebra, we can show that in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right] \approx 0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In the model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the 3 \(\times 3\) matrix. This number is low for two reasons. The first year of life is precarious for most animals living in the wild. In addition, juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, thanks to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Let \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right],\) where \(a, b,\) and \(c\) are nonzero real numbers. Find \(A^{-1}\). Let \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1\end{array}\right] .\) Show that \(A^{3}=I_{3},\) and use this result to find the inverse of \(A\).

$$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], \quad B=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right], \quad \text { and } \quad C=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] $$ where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A(B+C)=A B+A C\) (distributive property)
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