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Chapter 9: Problem 36

Perform each operation when possible. $$\left[\begin{array}{r} 4 k-8 y \\ 6 z-3 x \\ 2 k+5 a \\ -4 m+2 n \end{array}\right]-\left[\begin{array}{c} 5 k+6 y \\ 2 z+5 x \\ 4 k+6 a \\ 4 m-2 n \end{array}\right]$$

Short Answer

Expert verified
The result is \left[ \begin{array}{r} -k - 14y \ 4z - 8x \ -2k - a \ -8m + 4n \end{array}\right].

Step by step solution

01

- Set Up the Problem

Write down both matrices for clarity:\[A = \left[ \begin{array}{r} 4k-8y \ 6z-3x \ 2k+5a \ -4m+2n \end{array}\right] , \quad B = \left[ \begin{array}{c} 5k+6y \ 2z+5x \ 4k+6a \ 4m-2n \end{array} \right]\]
02

- Subtract Corresponding Entries

Subtract each element of matrix B from each corresponding element in matrix A:\[\left[ \begin{array}{r} (4k-8y)-(5k+6y) \ (6z-3x)-(2z+5x) \ (2k+5a)-(4k+6a) \ (-4m+2n)-(4m-2n) \end{array}\right]\]
03

- Simplify Each Expression

Combine like terms for each element:\[\left[ \begin{array}{r} 4k - 5k - 8y - 6y \ 6z - 2z - 3x - 5x \ 2k - 4k + 5a - 6a \ -4m - 4m + 2n - (-2n) \end{array}\right] = \left[ \begin{array}{r} -k - 14y \ 4z - 8x \ -2k - a \ -8m + 4n \end{array}\right]\]

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Key Concepts

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

matrix operations
Matrices are a fundamental concept in mathematics and are used to represent and solve various problems. Matrix operations, such as addition, subtraction, multiplication, and division, allow us to manipulate these structures to find solutions efficiently. In this exercise, we look at matrix subtraction which involves working with each element of the matrices.

Matrices can be added or subtracted when they have the same dimensions. For instance, we can only subtract two matrices if they both have the same number of rows and columns. This consistency is crucial for the matrix operations to make sense and yield meaningful results.

Understanding matrix operations is essential for more advanced topics in mathematics and related fields, such as linear algebra, physics, computer science, and engineering.
combining like terms
Combining like terms is a key algebraic process that simplifies expressions. In the context of matrices, this process helps us streamline and solve elements effectively during operations like addition and subtraction.

When we perform matrix subtraction, each element is a small algebraic expression. To simplify these expressions, we must combine like terms. For example, when subtracting \(5k + 6y\) from \(4k - 8y\), we combine terms involving \(k\) and \(y\) separately: \(4k - 5k - 8y - 6y \rightarrow -k - 14y\).

Understanding how to combine like terms makes it easier to simplify and solve complex expressions. This technique is not just limited to matrix operations but is widely used in algebra and calculus.
element-wise subtraction
Element-wise subtraction means subtracting each element of one matrix from the corresponding element of another matrix. This method is straightforward but requires careful attention to detail.

In our exercise, we first list both matrices and then subtract each element of matrix B from matrix A. The key steps are:
  • Subtract corresponding elements: \[ (4k - 8y) - (5k + 6y) \]
  • Simplify by combining like terms: \[ -k - 14y \]
Repeating this for each element, we obtain a new matrix.

Element-wise operations are essential for simplifying matrix expressions and solving matrix equations. Mastering these operations enables you to tackle more advanced mathematical problems and applications efficiently.
precalculus
Precalculus explores mathematical concepts that prepare students for calculus, including functions, complex numbers, and matrices. Matrix operations, such as the one in this exercise, form a crucial part of precalculus.

Understanding matrices and their operations lay a firm groundwork for calculus topics like integrating functions involving matrices, transformations, and eigenvalues. A solid grasp of matrix subtraction, multiplication, and addition is essential.

Precalculus also emphasizes critical problem-solving skills and logical reasoning, which are indispensable when dealing with matrices and higher-level math. Ensuring proficiency in these foundational areas helps students transition smoothly into calculus and beyond.

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Most popular questions from this chapter

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.

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{r} 2 x-y+3 z=1 \\ -2 x+y-3 z=2 \\ 5 x-y+z=2 \end{array}$$

Solve each problem. Plate-Glass Sales The amount of plate-glass sales \(S\) (in millions of dollars) can be affected by the number of new building contracts \(B\) issued (in millions) and automobiles \(A\) produced (in millions). A plate-glass company in California wants to forecast future sales by using the past three years of sales. The totals for the three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b A+c B $$. where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Makridakis, S., and S. Wheelwright, Forecasting Methods for Management, John Wiley and Sons.) (a) Substitute the values for \(S, A,\) and \(B\) for each year from the table into the equation \(S=a+b A+c B,\) 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. $$\begin{array}{|c|c|c|} \hline S & A & B \\ \hline 602.7 & 5.543 & 37.14 \\\ \hline 656.7 & 6.933 & 41.30 \\ \hline 778.5 & 7.638 & 45.62 \\ \hline \end{array}$$ (d) For the next year it is estimated that \(A=7.752\) and \(B=47.38 .\) Predict \(S .\) (The actual value for \(S\) was \(877.6 .\) ) (e) It is predicted that in 6 yr, \(A=8.9\) and \(B=66.25 .\) Find the value of \(S\) in this situation and discuss its validity.

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

Concept Check Find \(A B\) and \(B A\) for $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$ What do you notice? Matrix \(B\) acts as the multiplicative ___________ element for \(2 \times 2\) square matrices.
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