Use the given row transformation to change each matrix as indicated. [ 1 − 4 7 0 ] , − 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}

Hello, everyone. We are asked to apply the indicated row transformation to the following matrix. We are given a two by two matrix where the first row is six negative nine. And the second row is 11 0, we are asked to add the result negative 10 times row one to row two. We have four answer choices all of which are two by two matrices with slightly different answers in them. So thinking about what add the result negative 10 times of row one to row two means this means we're gonna do negative 10 multiplied by the element from row one. And then we're gonna add it to the element from row two. So in doing so row one does not change. So it remains six negative nine. However, in row two, we need to do negative 10 multiplied by the corresponding value from row one to the first element of row one, which is six. And then we're gonna add the element from row two in our original matrix which is 11. Same thing with the second element for row two. We're gonna do negative 10 multiplied by the element in the second spot in row one. Which is negative nine and add the element that is originally in row two, which is zero. So now doing out the multiplication in addition, again, row one is not changing. It is still six negative nine row two. The first element negative 10 multiplied by six is negative 60 negative 60 plus 11 is negative 49. And in the second element of row two, negative 10 multiplied by negative nine is positive 90 plus zero is still positive 90. So our answer is a two by two matrix where the top row is six negative nine. And the second row is negative 49 90. And this matches answer choice A have a nice day.

7. Systems of Equations & Matrices

Introduction to Matrices

College Algebra

7. Systems of Equations & Matrices

Introduction to Matrices

2:37 minutes

Problem 8

Textbook Question

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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