Use the given row transformation to change each matrix as indicated. [1−470],−7×row 1 added to row 2\left[ \begin{matrix} 1 & -4 \\ 7 & 0 \end{matrix} \right], \quad -7 \times \text{row 1 added to row 2}

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 8

Chapter 6, Problem 8

Use the given row transformation to change each matrix as indicated.
$[\begin{matrix} 1 & - 4 \\ 7 & 0 \end{matrix}],$

Verified step by step guidance

Identify the given matrix as a 2x2 matrix, which can be represented as $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, where $a$, $b$, $c$, and $d$ are the elements of the matrix.

Understand the row operation: '-7 times row 1 added to row 2' means you multiply each element of row 1 by -7 and then add the result to the corresponding element in row 2.

Write the row operation mathematically as: $R_2 \rightarrow R_2 + (-7) \times R_1$, where $R_1$ and $R_2$ represent row 1 and row 2 respectively.

Apply the operation to each element in row 2: calculate the new element in row 2, column 1 as $c + (-7) \times a$, and the new element in row 2, column 2 as $d + (-7) \times b$.

Construct the new matrix after the row operation, keeping row 1 unchanged and replacing row 2 with the new calculated elements.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Elementary Row Operations

Elementary row operations are basic manipulations performed on the rows of a matrix to simplify or solve systems of equations. These include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. Understanding these operations is essential for matrix transformations and solving linear systems.

Matrix Representation and Notation

A matrix is a rectangular array of numbers arranged in rows and columns. Each element is identified by its row and column position. Recognizing how to read and write matrices, especially 2x2 matrices, is crucial for applying row operations correctly and interpreting the results.

Row Transformation: Adding a Multiple of One Row to Another

This specific row operation involves multiplying one row by a scalar and adding it to another row, replacing the latter. For example, '-7 times row 1 added to row 2' means multiply row 1 by -7 and add it to row 2. This operation helps in creating zeros or simplifying matrices during row reduction.

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Related Practice

Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices.

$[\begin{matrix} - 6 & 8 & 0 & 0 \\ 4 & 1 & 9 & 2 \\ 3 & - 5 & 7 & 1 \end{matrix}]$

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Textbook Question

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.

6x + 10y = -11

9x + 6y = -3

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Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices.

$[\begin{matrix} - 9 & 6 & 2 \\ 4 & 1 & 8 \end{matrix}]$

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Textbook Question

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations.

2x² = 3y + 23

y = 2x - 5

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Textbook Question

Evaluate each determinant.

$∣ \begin{matrix} - 1 & - 2 \\ 5 & 3 \end{matrix} ∣$