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Chapter 6: Problem 39

Find matrix \(A\) if $$B=\left[\begin{array}{rrr} 4 & 6 & -5 \\ -6 & 3 & 2 \end{array}\right] \text { and } A+B=\left[\begin{array}{rrr} 6 & 12 & 0 \\ -10 & -4 & 11 \end{array}\right].$$

Short Answer

Expert verified
Matrix \( A = \begin{bmatrix} 2 & 6 & 5 \\ -4 & -7 & 9 \end{bmatrix} \).

Step by step solution

01

Understanding the Problem

We need to find matrix \( A \) given the matrix \( B \) and the equation \( A + B = C \), where \( C \) is already defined. Matrix addition means the sum of corresponding elements, so we can solve for each element of matrix \( A \).
02

Identifying Known Matrices

Matrix \( B \) is given by \[B = \begin{bmatrix} 4 & 6 & -5 \ -6 & 3 & 2 \end{bmatrix}\] and matrix \( C \) is given by \[C = A + B = \begin{bmatrix} 6 & 12 & 0 \ -10 & -4 & 11 \end{bmatrix}.\] We need to solve for \( A \).
03

Setting Up the Equation

Using the equation \( A + B = C \), we can write the equation to find \( A \) as: \[ A = C - B \].
04

Subtraction of Matrices

To find \( A \), we subtract each element of \( B \) from \( C \):\[A = \begin{bmatrix} 6 - 4 & 12 - 6 & 0 - (-5) \ -10 - (-6) & -4 - 3 & 11 - 2 \end{bmatrix}\] which simplifies to: \[A = \begin{bmatrix} 2 & 6 & 5 \ -4 & -7 & 9 \end{bmatrix}\]
05

Verify the Solution

Add matrices \( A \) and \( B \) to verify if they equal \( C \). Calculate: \[A + B = \begin{bmatrix} 2 + 4 & 6 + 6 & 5 - 5 \ -4 - 6 & -7 + 3 & 9 + 2 \end{bmatrix} = \begin{bmatrix} 6 & 12 & 0 \ -10 & -4 & 11 \end{bmatrix}\] which matches the given matrix \( C \).

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Key Concepts

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

Matrix Addition
Matrix addition is a fundamental concept in matrix algebra. It involves adding corresponding elements of two matrices to form a new matrix. To perform matrix addition, you need two matrices that have the exact same dimensions, meaning they must have the same number of rows and columns.
For example, when adding matrices \( A \) and \( B \), the resultant matrix \( C \) is determined by calculating \( A_{ij} + B_{ij} = C_{ij} \), where \( i \) is the row index and \( j \) is the column index.
  • Matrix \( A \) is added to matrix \( B \) by adding each element \( A[i][j] \) with \( B[i][j] \).
  • The result is a new matrix of the same dimensions as the original matrices.
  • This process is only feasible if both matrices share identical dimensions.
In the original exercise, we used this principle in reverse to verify our solution, by checking the sums after performing matrix subtraction to find \( A \).
Matrix Subtraction
Matrix subtraction is the process of subtracting each element of one matrix from the corresponding element of another matrix. Just like addition, this can only occur between matrices of the same size.
We express matrix subtraction for matrices \( A \) and \( B \) as \( C = A - B \). This means \( C[i,j] = A[i,j] - B[i,j] \).
  • Each element of the resulting matrix \( C \) is computed by subtracting the element in matrix \( B \) from the corresponding element in matrix \( A \).
  • The dimensions of matrices \( A \) and \( B \) must match to perform subtraction.
  • Matrix subtraction is often used to solve problems involving matrix equations, like finding unknown matrices.
In the original exercise, we used subtraction to solve for matrix \( A \) by applying the equation \( A = C - B \). This allowed us to determine the unknown matrix \( A \) by handling each element individually.
Matrix Equations
Matrix equations are similar to algebraic equations but involve matrices instead of numbers. A matrix equation such as \( A + B = C \) requires consistent operations on matrices to find the unknown matrix.
The principles of solving matrix equations involve ensuring that operations are valid, like having the correct dimensions for matrix addition or subtraction.
  • To solve \( A + B = C \) for an unknown matrix \( A \), you can rearrange it to \( A = C - B \).
  • Use matrix subtraction rules to find each element of the unknown matrix.
  • Once the unknown matrix is found, verify the solution by substituting back into the original equation.
This exercise demonstrated solving matrix equations by isolating the variable matrix using subtraction and then verifying through addition.
Elementary Row Operations
Elementary row operations are a set of techniques used to manipulate matrices, often for solving linear systems or performing row reduction.
The three primary row operations include:
  • Swapping two rows.
  • Multiplying a row by a non-zero constant.
  • Adding or subtracting a multiple of one row from another row.
These operations are vital in achieving row-echelon form for solving systems of linear equations or finding determinants.
While our exercise was more focused on matrix arithmetic, understanding elementary row operations is crucial for more advanced matrix manipulations and can simplify complex matrix problems.

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Most popular questions from this chapter

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years 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. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this 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, and 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, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Given a square matrix \(A^{-1}\), find matrix \(A\). 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 \(A\).

Solve each system of four equations in four variables. Express the solutions in the form \((x, y, z, w)\) $$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\\ x-y-3 z-2 w &=2 \end{aligned}$$

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{aligned}&\pi x+e y=3\\\&e x+\pi y=4\end{aligned}$$

The relationship between a professional basketball player's height \(h\) in inches and weight \(w\) in pounds was modeled by using two samples of players. The resulting equations were $$\begin{aligned}&w=7.46 h-374\\\&w=7.93 h-405\end{aligned}$$ and Assume that \(65 \leq h \leq 85\) (a) Use each equation to predict the weight to the nearest pound of a professional basketball player who is 6 feet 11 inches. (b) Determine graphically the height at which the two models give the same weight. (c) For each model, what change in weight is associated with a 1 -inch increase in height?
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