Apply The Row Operations { R_2 = -4R_1 + R_2 $}$ And { R_3 = 7R_1 + R_3 $}$ On The Matrix:$[ \left[\begin{array}{rrr|r} 1 & -7 & 4 & 4 \ 4 & -6 & 6 & 3 \ -7 & 5 & 4 &

In linear algebra, row operations are a set of techniques used to manipulate the rows of a matrix. These operations are essential in solving systems of linear equations and finding the inverse of a matrix. In this article, we will apply two row operations to a given matrix and explore the effects of these operations on the matrix.

The Given Matrix

The given matrix is:

$${ \left[\begin{array}{rrr|r} 1 & -7 & 4 & 4 \\ 4 & -6 & 6 & 3 \\ -7 & 5 & 4 & 2 \\ \end{array}\right] }$$

Applying the First Row Operation

The first row operation is:

$${ R_2 = -4R_1 + R_2 }$$

This operation involves multiplying the first row by -4 and adding the result to the second row. To apply this operation, we will multiply each element of the first row by -4 and add the corresponding elements of the second row.

Step 1: Multiply the First Row by -4

The first row is:

$${ \left[\begin{array}{rrr|r} 1 & -7 & 4 & 4 \\ \end{array}\right] }$$

Multiplying this row by -4 gives:

$${ \left[\begin{array}{rrr|r} -4 & 28 & -16 & -16 \\ \end{array}\right] }$$

Step 2: Add the Corresponding Elements

The second row is:

$${ \left[\begin{array}{rrr|r} 4 & -6 & 6 & 3 \\ \end{array}\right] }$$

Adding the corresponding elements of the two rows gives:

$${ \left[\begin{array}{rrr|r} -4+4 & 28-6 & -16+6 & -16+3 \\ \end{array}\right] }$$

Simplifying the result gives:

$${ \left[\begin{array}{rrr|r} 0 & 22 & -10 & -13 \\ \end{array}\right] }$$

The Result of the First Row Operation

The result of the first row operation is:

$${ \left[\begin{array}{rrr|r} 1 & -7 & 4 & 4 \\ 0 & 22 & -10 & -13 \\ -7 & 5 & 4 & 2 \\ \end{array}\right] }$$

Applying the Second Row Operation

The second row operation is:

$${ R_3 = 7R_1 + R_3 }$$

This operation involves multiplying the first row by 7 and adding the result to the third row. To apply this operation, we will multiply each element of the first row by 7 and add the corresponding elements of the third row.

Step 1: Multiply the First Row by 7

The first row is:

$${ \left[\begin{array}{rrr|r} 1 & -7 & 4 & 4 \\ \end{array}\right] }$$

Multiplying this row by 7 gives:

$${ \left[\begin{array}{rrr|r} 7 & -49 & 28 & 28 \\ \end{array}\right] }$$

Step 2: Add the Corresponding Elements

The third row is:

$${ \left[\begin{array}{rrr|r} -7 & 5 & 4 & 2 \\ \end{array}\right] }$$

Adding the corresponding elements of the two rows gives:

$${ \left[\begin{array}{rrr|r} 7-7 & -49+5 & 28+4 & 28+2 \\ \end{array}\right] }$$

Simplifying the result gives:

$${ \left[\begin{array}{rrr|r} 0 & -44 & 32 & 30 \\ \end{array}\right] }$$

The Result of the Second Row Operation

The result of the second row operation is:

$${ \left[\begin{array}{rrr|r} 1 & -7 & 4 & 4 \\ 0 & 22 & -10 & -13 \\ 0 & -44 & 32 & 30 \\ \end{array}\right] }$$

Conclusion

In this article, we applied two row operations to a given matrix. The first row operation involved multiplying the first row by -4 and adding the result to the second row. The second row operation involved multiplying the first row by 7 and adding the result to the third row. The results of these operations are essential in solving systems of linear equations and finding the inverse of a matrix.

Discussion

Row operations are a fundamental concept in linear algebra. They are used to manipulate the rows of a matrix and are essential in solving systems of linear equations. In this article, we applied two row operations to a given matrix and explored the effects of these operations on the matrix.

Further Reading

For further reading on row operations and linear algebra, we recommend the following resources:

• Linear Algebra and Its Applications by Gilbert Strang
• Introduction to Linear Algebra by David C. Lay
• Linear Algebra: A Modern Introduction by David Poole

In this article, we will answer some frequently asked questions on row operations, including their definition, importance, and applications.

Q: What are row operations?

A: Row operations are a set of techniques used to manipulate the rows of a matrix. These operations involve adding, subtracting, or multiplying rows by scalars, and are essential in solving systems of linear equations and finding the inverse of a matrix.

Q: Why are row operations important?

A: Row operations are important because they allow us to transform a matrix into a simpler form, making it easier to solve systems of linear equations. They are also essential in finding the inverse of a matrix, which is used in many applications, including computer graphics, data analysis, and machine learning.

Q: What are the different types of row operations?

A: There are three main types of row operations:

1. Row addition: Adding a multiple of one row to another row.
2. Row subtraction: Subtracting a multiple of one row from another row.
3. Row multiplication: Multiplying a row by a scalar.

Q: How do I apply row operations to a matrix?

A: To apply row operations to a matrix, you need to follow these steps:

1. Identify the row operation: Determine which row operation you want to apply, such as adding or subtracting a multiple of one row from another row.
2. Calculate the new row: Calculate the new row by applying the row operation to the original row.
3. Replace the original row: Replace the original row with the new row.

Q: What are some common mistakes to avoid when applying row operations?

A: Some common mistakes to avoid when applying row operations include:

1. Incorrectly calculating the new row: Make sure to calculate the new row correctly by applying the row operation to the original row.
2. Not replacing the original row: Make sure to replace the original row with the new row after applying the row operation.
3. Not checking for row dependencies: Make sure to check for row dependencies before applying row operations.

Q: How do I know when to stop applying row operations?

A: You can stop applying row operations when:

1. The matrix is in row echelon form: The matrix is in row echelon form when all the rows above the leading entry of each row are zero.
2. The matrix is in reduced row echelon form: The matrix is in reduced row echelon form when all the rows above the leading entry of each row are zero, and the leading entry of each row is 1.

Q: What are some real-world applications of row operations?

A: Row operations have many real-world applications, including:

1. Computer graphics: Row operations are used in computer graphics to transform 2D and 3D objects.
2. Data analysis: Row operations are used in data analysis to transform and manipulate data.
3. Machine learning: Row operations are used in machine learning to transform and manipulate data.

Conclusion

In this article, we answered some frequently asked questions on row operations, including their definition, importance, and applications. We also provided some tips and tricks for applying row operations correctly and avoiding common mistakes.