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Chapter 7: Problem 29

Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{rr}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(100 D-10 E\)

Short Answer

Expert verified
The result of \( 100D - 10E \) is \( \left[\begin{array}{rrr}-840 & 650 & -530 \\ 330 & 360 & 250 \\ -10 & 900 & 110\end{array}\right] \).

Step by step solution

01

Multiply Matrix D by 100

Matrix D is \( \left[\begin{array}{rrr}-8 & 7 & -5 \ 4 & 3 & 2 \ 0 & 9 & 2\end{array}\right] \). We will multiply every element of Matrix D by 100. This gives us: \[100 \times D = \left[\begin{array}{rrr}-800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200\end{array}\right]\]
02

Multiply Matrix E by 10

Matrix E is \( \left[\begin{array}{rrr}4 & 5 & 3 \ 7 & -6 & -5 \ 1 & 0 & 9\end{array}\right] \). We will multiply every element of Matrix E by 10. This gives us: \[10 \times E = \left[\begin{array}{rrr}40 & 50 & 30 \ 70 & -60 & -50 \ 10 & 0 & 90\end{array}\right] \]
03

Subtract 10E from 100D

We need to perform the subtraction \( 100D - 10E \) using the matrices from Steps 1 and 2.\[100D - 10E = \left[\begin{array}{rrr}-800 & 700 & -500 \ 400 & 300 & 200 \ 0 & 900 & 200\end{array}\right] - \left[\begin{array}{rrr}40 & 50 & 30 \ 70 & -60 & -50 \ 10 & 0 & 90\end{array}\right]\]Subtract each corresponding element:- First row: \(-800 - 40 = -840\), \(700 - 50 = 650\), \(-500 - 30 = -530\)- Second row: \(400 - 70 = 330\), \(300 - (-60) = 360\), \(200 - (-50) = 250\)- Third row: \(0 - 10 = -10\), \(900 - 0 = 900\), \(200 - 90 = 110\)Thus, \[100D - 10E = \left[\begin{array}{rrr}-840 & 650 & -530 \ 330 & 360 & 250 \ -10 & 900 & 110\end{array}\right]\]
04

Final Result

The resulting matrix from the operation \( 100D - 10E \) is \[\left[\begin{array}{rrr}-840 & 650 & -530 \ 330 & 360 & 250 \ -10 & 900 & 110\end{array}\right]\].

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Key Concepts

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

Matrix Subtraction
Matrix subtraction is a fundamental concept in matrix operations. It involves taking two matrices of the same dimensions and subtracting their corresponding elements.
For matrices to be subtracted, they must have an equal number of rows and columns. This condition ensures that each element in one matrix has a corresponding element in the other.
To perform matrix subtraction:
  • Take each pair of corresponding elements from the two matrices.
  • Subtract the element in the second matrix from the element in the first matrix.
As seen in the example with the matrices resulting from multiplying matrices D and E, subtraction is done element by element to create a new matrix. If matrices have different sizes, matrix subtraction cannot be carried out, showing the importance of matching dimensions.
Scalar Multiplication of Matrices
Scalar multiplication involves multiplying every element of a matrix by a single number, known as a scalar. This operation scales the entire matrix by a constant factor.
The process of scalar multiplication can be summarized as:
  • Take the scalar value.
  • Multiply each element of the matrix by the scalar.
In the exercise, to compute \(100D\) and \(10E\), each element in matrix D was multiplied by 100, and each element in matrix E was multiplied by 10. This operation is straightforward and involves only basic arithmetic, making it one of the more accessible types of matrix operations.
Matrix Arithmetic
Matrix arithmetic includes operations such as addition, subtraction, and multiplication. These operations are essential for various applications in mathematics and related fields.
When handling matrix arithmetic:
  • Ensure matrices are compatible for operations by matching their dimensions.
  • Follow the rules of matrix operations consistently for accurate results.
  • Understand the properties of different operations. For instance, matrix addition and subtraction require matrices of the same size.
In this exercise, understanding scalar multiplication allowed for initial scaling of matrices D and E. Subsequent subtraction involved direct use of these modified matrices, demonstrating the power of combining basic operations for more complex arithmetic tasks in linear algebra.

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Most popular questions from this chapter

For the following exercises, use Gaussian elimination to solve the system. $$ \begin{aligned} \frac{x-3}{4}-\frac{y-1}{3}+2 z &=-1 \\ \frac{x+5}{2}+\frac{y+5}{2}+\frac{z+5}{2} &=7 \\ x+y+z &=1 \end{aligned} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} 4 x-6 y+8 z &=10 \\ -2 x+3 y-4 z &=-5 \\ 12 x+18 y-24 z &=-30 \end{aligned} $$

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer鈥檚 Rule. At the same market, the three most popular fruits make up 37% of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. \(30 \%\) of the almonds, \(20 \%\) of the cashews, and \(10 \%\) of the pistachios were eaten, and now there are 770 nuts left in he bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was 348 lb, 12 oz. If the green bean cans weigh 2 oz less than the kidney bean cans, how many of each can was donated?
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