Perform each matrix row operation and write the new matrix. [ 1 2 2 ∣ 2 0 1 − 1 ∣ 2 0 5 4 ∣ 1 ] − 5 R 2 + R 3 \begin{bmatrix}1 & 2 & 2 & \vert & 2 \\0 & 1 & -1 & \vert & 2 \\0 & 5 & 4 & \vert & 1\end{bmatrix}-5R_2 + R_3 1 0 0 2 1 5 2 − 1 4 ∣ ∣ ∣ 2 2 1 − 5 R 2 + R 3

Welcome back. I'm so glad you're here. We're going to write a new matrix. After applying the mentioned row operations, I'm going to scroll down a little bit so we can see everything and so we're going to look at this matrix that we've got here. The first row is 1 to 13, 2nd rose to 111. And the third row is 4 to 68. And the labels look like this are sub one for row one, our sub two for row two and our sub three for row three. And we want to Take row three and we're going to divide everything in there by two and then add the result of that two, row two. So let's just think about that. What is row three divided by two? Well four divided by two is two, two divided by two is 16 divided by two is three and eight divided by two is four. And now that's going to get added to row two. So that means Rose one and three aren't going to change. We can copy those down 1213 and Rose three stays the same. 4268. Now let's just add our red numbers. The row three divided by two to row two, so two plus two is 41 plus one is two, three plus one is four, and four plus one is five. We look at our answer choices and that is going to match answer choice. C Well then we'll catch you on the next one

7. Systems of Equations & Matrices

Introduction to Matrices

College Algebra

7. Systems of Equations & Matrices

Introduction to Matrices

2:00 minutes

Problem 1

Textbook Question

Perform each matrix row operation and write the new matrix.
$\begin{bmatrix}1 & 2 & 2 & \vert & 2 \\0 & 1 & -1 & \vert & 2 \\0 & 5 & 4 & \vert & 1\end{bmatrix}-5R_2 + R_3$

Verified step by step guidance

Identify the specific matrix and the row operation to be performed. Common row operations include swapping two rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another row.

Write down the original matrix clearly, labeling each row (e.g., R1, R2, R3, etc.) to keep track of changes.

Apply the given row operation step-by-step. For example, if the operation is to add 3 times row 1 to row 2, calculate each element of the new row 2 by performing the operation element-wise.

Replace the affected row in the original matrix with the new row obtained from the operation, keeping the other rows unchanged.

Write the resulting matrix after the row operation, ensuring all entries are correctly updated and the matrix is clearly presented.

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Key Concepts

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

Matrix Row Operations

Matrix row operations are fundamental manipulations applied to the rows of a matrix to simplify or solve systems of equations. These include swapping two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. Understanding these operations is essential for transforming matrices into simpler forms.

Elementary Row Operations and Their Effects

Elementary row operations change the matrix while preserving the solution set of the corresponding system. Each operation corresponds to an invertible transformation, allowing the matrix to be converted into row echelon or reduced row echelon form, which aids in solving linear systems or finding matrix inverses.

Matrix Notation and Representation

A matrix is a rectangular array of numbers arranged in rows and columns. Understanding how to read and write matrices, including identifying rows and columns, is crucial for correctly performing and recording row operations. Proper notation ensures clarity when showing the steps of matrix manipulation.

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