https://file.aiquickdraw.com/mathbot/file/1727576833978yn2ysegu.jpg

The given image shows a matrix with a set of elementary row operations indicated. Let's break it down:

The matrix given is:

\[
\begin{bmatrix}
2 & 6 & 4 & -6 \\
1 & 3 & 2 & -3 \\
1 & -1 & 0 & -2
\end{bmatrix}
\]

The indicated row operations are:

1. \(-2R_2 + R_1\)
2. \(-4R_2 + R_3\)

### First row operation: \(-2R_2 + R_1\)

We calculate each element of the new \(R_1\) by multiplying row \(R_2\) by \(-2\) and adding the result to row \(R_1\).

For each element in \(R_1\):

\[
R_1 \text{ new} = R_1 + (-2)R_2
\]

- First element: \( 2 + (-2)(1) = 2 - 2 = 0 \)
- Second element: \( 6 + (-2)(3) = 6 - 6 = 0 \)
- Third element: \( 4 + (-2)(2) = 4 - 4 = 0 \)
- Fourth element: \( -6 + (-2)(-3) = -6 + 6 = 0 \)

So, the new \(R_1\) is:

\[
R_1 = \begin{bmatrix} 0 & 0 & 0 & 0 \end{bmatrix}
\]

### Second row operation: \(-4R_2 + R_3\)

We calculate each element of the new \(R_3\) by multiplying row \(R_2\) by \(-4\) and adding the result to row \(R_3\).

For each element in \(R_3\):

\[
R_3 \text{ new} = R_3 + (-4)R_2
\]

- First element: \( 1 + (-4)(1) = 1 - 4 = -3 \)
- Second element: \( -1 + (-4)(3) = -1 - 12 = -13 \)
- Third element: \( 0 + (-4)(2) = 0 - 8 = -8 \)
- Fourth element: \( -2 + (-4)(-3) = -2 + 12 = 10 \)

So, the new \(R_3\) is:

\[
R_3 = \begin{bmatrix} -3 & -13 & -8 & 10 \end{bmatrix}
\]

### Updated Matrix:

After performing the row operations, the new matrix is:

\[
\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 & 3 & 2 & -3 \\
-3 & -13 & -8 & 10
\end{bmatrix}
\]

Let me know if you need further details or explanations!

Here are five related questions:

1. How do we perform the Gaussian elimination process?
2. What are the different types of elementary row operations?
3. How does row reduction help solve systems of linear equations?
4. Can a matrix be reduced to row echelon form using elementary row operations?
5. How do we interpret matrices with rows of all zeros?

**Tip:** Always keep track of the row operations applied in each step to avoid mistakes in further operations.

Fill in the blank by performing the indicated elementary row operation(s). The initial matrix is: 
[[2, 6, 4, -6], 
[1, 3, 2, -3], 
[1, -1, 0, -2]]. The operations to perform are: -2R2 + R1, -4R2 + R3.

Perform elementary row operations on a 3x4 matrix by applying the operations -2R2 + R1 and -4R2 + R3. Learn how to simplify matrices through these transformations, essential for solving systems of linear equations. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation