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Chapter 8: Problem 22

Let \(A=\left[\begin{array}{lll}a & b & c \\ d & e & f\end{array}\right] \quad E_{1}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \quad E_{2}=\left[\begin{array}{ll}5 & 0 \\ 0 & 1\end{array}\right] \quad E_{3}=\left[\begin{array}{lr}1 & -2 \\ 0 & 1\end{array}\right]\) Compute \(E_{1} A, \quad E_{2} A\) and \(E_{3} A .\) What effect did each of the \(E_{i}\) matrices have on the rows of \(A ?\) Create \(E_{4}\) so that its effect on \(A\) is to multiply the bottom row by \(-6 .\) How would you extend this idea to matrices with more than two rows?

Short Answer

Expert verified
Matrix \(E_1\) swaps rows, \(E_2\) multiplies the first row by 5, and \(E_3\) subtracts twice the second row from the first. Matrix \(E_4\) multiplies the second row by -6.

Step by step solution

01

Compute Matrix Multiplication for E1A

The matrix multiplication of two matrices is performed by taking the dot product of rows and columns from the first and second matrix, respectively. For matrix \(E_1=\begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix}\) and matrix \(A=\begin{pmatrix}a & b & c \ d & e & f\end{pmatrix}\), we compute:\[E_1 \cdot A = \begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix} \begin{pmatrix}a & b & c \ d & e & f\end{pmatrix} = \begin{pmatrix}d & e & f \ a & b & c\end{pmatrix}\] The effect of matrix \(E_1\) is that it swaps the rows of matrix \(A\).
02

Compute Matrix Multiplication for E2A

Using the given matrix \(E_2 = \begin{pmatrix}5 & 0 \ 0 & 1\end{pmatrix}\) and matrix \(A\):\[E_2 \cdot A = \begin{pmatrix}5 & 0 \ 0 & 1\end{pmatrix} \begin{pmatrix}a & b & c \ d & e & f\end{pmatrix} = \begin{pmatrix}5a & 5b & 5c \ d & e & f\end{pmatrix}\] The effect of matrix \(E_2\) is to multiply the entire first row of matrix \(A\) by 5.
03

Compute Matrix Multiplication for E3A

Given matrix \(E_3 = \begin{pmatrix}1 & -2 \ 0 & 1\end{pmatrix}\):\[E_3 \cdot A = \begin{pmatrix}1 & -2 \ 0 & 1\end{pmatrix} \begin{pmatrix}a & b & c \ d & e & f\end{pmatrix} = \begin{pmatrix}a - 2d & b - 2e & c - 2f \ d & e & f\end{pmatrix}\] The effect of matrix \(E_3\) is to subtract twice the second row from the first row of matrix \(A\).
04

Create E4 to Multiply Bottom Row by -6

To multiply the bottom row of \(A\) by -6, we consider matrix \(E_4\):\[E_4 = \begin{pmatrix}1 & 0 \ 0 & -6\end{pmatrix}\]The multiplication \(E_4 \cdot A\) will yield:\[\begin{pmatrix}1 & 0 \ 0 & -6\end{pmatrix} \begin{pmatrix}a & b & c \ d & e & f\end{pmatrix} = \begin{pmatrix}a & b & c \ -6d & -6e & -6f\end{pmatrix}\] This matrix \(E_4\) will multiply the second row of matrix \(A\) by -6.
05

Extending to More than Two Rows

For a matrix \(A\) with more than two rows, an elementary matrix \(E\) that affects a specific row \(i\) while leaving others unchanged can be structured as an identity matrix, except with the \(i^{th}\) diagonal entry modified as needed. To apply an operation to a row, insert the scaler in place of the diagonal entry corresponding to the row to modify. For example, to multiply the third row by a constant \(c\), use \(I\) with \(c\) at position \((3, 3)\).

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Key Concepts

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

Elementary Matrices
Elementary matrices are fundamental tools in linear algebra that help us perform transformations on other matrices easily. An elementary matrix is constructed by applying a single row operation to an identity matrix. These matrices are crucial because they help in simplifying matrices, finding inverses, and solving linear equations.
  • Elementary matrices are of the same size as the matrix they transform.
  • They are derived from identity matrices, which have ones on the diagonal and zeros elsewhere.
  • A single row operation on an identity matrix results in an elementary matrix.
For instance, when you multiply an elementary matrix with another matrix, it replicates the same row operation on that matrix, simplifying processes related to matrix equations.
Row Operations
Row operations are essential manipulations used to solve systems of linear equations and bring matrices to different forms like echelon or reduced echelon forms. There are three main types of row operations:
  • Swapping two rows.
  • Multiplying a row by a non-zero scalar.
  • Adding or subtracting a multiple of one row to another.
These operations do not change the solutions of the system represented by the matrix, making them a critical tool in linear algebra. Their formal application is realized through multiplying by elementary matrices. Each operation corresponds to a specific type of elementary matrix.
Matrix Row Swapping
Matrix row swapping is a simple yet powerful row operation where two rows of a matrix are exchanged. This operation can be extremely helpful in solving linear systems and adjusting matrix forms.
  • To swap two rows in a matrix, use an elementary matrix where two of its row entries are swapped.
  • For matrix \(A\), if you use matrix \(E_1\) with form \(\begin{pmatrix}0 & 1 \ 1 & 0\end{pmatrix}\), it swaps the first and second rows.
  • This operation is often used to bring a pivot to the top of a column or to simplify further operations in solving linear systems.
Matrix row swapping, although simple, is a fundamental operation that can lead to more efficient calculations.
Matrix Scaling
Matrix scaling refers to multiplying all elements of a particular row within a matrix by a scalar. This operation is crucial for adjusting certain rows without affecting the rest of the matrix. It is particularly useful in normalizing rows or setting up matrices for further operations.
  • To scale a row of matrix \(A\) by a scalar \(k\), use an elementary matrix derived from an identity matrix with \(k\) replacing the diagonal element of the specific row.
  • For instance, multiplying the top row by 5 in matrix \(A\) involves using matrix \(E_2\) with the form \(\begin{pmatrix}5 & 0 \ 0 & 1\end{pmatrix}\).
  • Matrix scaling is an operation that assists in matrix normalization and is a routine step in processes like Gaussian elimination.
Understanding matrix scaling and its applications allows for precise control over row operations in more complex matrix solutions.

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Most popular questions from this chapter

Solve the given system of nonlinear equations. Use a graph to help you avoid any potential extraneous solutions. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ y-x &=1 \end{aligned}\right. $$

Factor \(q(x)=x^{4}+6 x^{2}-5 x+6\).

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{m}{(7 x-6)\left(x^{2}+9\right)} $$

In Exercises \(23-29,\) consider the following scenario. In the small village of Pedimaxus in the country of Sasquatchia, all 150 residents get one of the two local newspapers. Market research has shown that in any given week, \(90 \%\) of those who subscribe to the Pedimaxus Tribune want to keep getting it, but \(10 \%\) want to switch to the Sasquatchia Picayune. Of those who receive the Picayune, \(80 \%\) want to continue with it and \(20 \%\) want switch to the Tribune. We can express this situation using matrices. Specifically, let \(X\) be the 'state matrix' given by $$ X=\left[\begin{array}{l} T \\ P \end{array}\right] $$ where \(T\) is the number of people who get the Tribune and \(P\) is the number of people who get the Picayune in a given week. Let \(Q\) be the 'transition matrix' given by $$ Q=\left[\begin{array}{ll} 0.90 & 0.20 \\ 0.10 & 0.80 \end{array}\right] $$ such that \(Q X\) will be the state matrix for the next week. Let's assume that when Pedimaxus was founded, all 150 residents got the Tribune. (Let's call this Week \(0 .)\) This would mean $$ X=\left[\begin{array}{r} 150 \\ 0 \end{array}\right] $$ Since \(10 \%\) of that 150 want to switch to the Picayune, we should have that for Week 1, 135 people get the Tribune and 15 people get the Picayune. Show that \(Q X\) in this situation is indeed $$ Q X=\left[\begin{array}{r} 135 \\ 15 \end{array}\right] $$

A Sasquatch's diet consists of three primary foods: Ippizuti Fish, Misty Mushrooms, and Sun Berries. Each serving of Ippizuti Fish is 500 calories, contains 40 grams of protein, and has no Vitamin X. Each serving of Misty Mushrooms is 50 calories, contains 1 gram of protein, and 5 milligrams of Vitamin X. Finally, each serving of Sun Berries is 80 calories, contains no protein, but has 15 milligrams of Vitamin \(\mathrm{X} .{ }^{9}\) (a) If an adult male Sasquatch requires 3200 calories, 130 grams of protein, and 275 milligrams of Vitamin X daily, use a matrix inverse to find how many servings each of Ippizuti Fish, Misty Mushrooms, and Sun Berries he needs to eat each day. (b) An adult female Sasquatch requires 3100 calories, 120 grams of protein, and 300 milligrams of Vitamin X daily. Use the matrix inverse you found in part (a) to find how many servings each of Ippizuti Fish, Misty Mushrooms, and Sun Berries she needs to eat each day. (c) An adolescent Sasquatch requires 5000 calories, 400 grams of protein daily, but no Vita\(\min \mathrm{X}\) daily. \({ }^{10}\) Use the matrix inverse you found in part (a) to find how many servings each of Ippizuti Fish, Misty Mushrooms, and Sun Berries she needs to eat each day.
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