Perform Each Matrix Row Operation And Write The New Matrix.Starting Matrix:$\[ \left[ \begin{array}{rrrr|r} 1 & -1 & -1 & 1 & 3 \\ 0 & 1 & -3 & -6 & 0 \\ 2 & 0 & -1 & 5 & 12 \\ 7 & 4 & 3 & 6 & 5 \end{array} \right] \\]Operation: \[$-2R_1 +

Introduction

In linear algebra, matrix row operations are a crucial tool for solving systems of linear equations and manipulating matrices. These operations involve adding or subtracting multiples of one row from another row, or multiplying a row by a scalar. In this article, we will perform a specific matrix row operation on a given starting matrix and write the new matrix.

Starting Matrix

The starting matrix is given as:

{ \left[ \begin{array}{rrrr|r} 1 & -1 & -1 & 1 & 3 \\ 0 & 1 & -3 & -6 & 0 \\ 2 & 0 & -1 & 5 & 12 \\ 7 & 4 & 3 & 6 & 5 \end{array} \right] \}

Operation: $-2R_1 + R_2$

The operation to be performed is $-2R_1 + R_2$. This means that we will multiply the first row ($R_1$) by $-2$ and add the result to the second row ($R_2$).

Step 1: Multiply the First Row by $-2$

To multiply the first row by $-2$, we will multiply each element of the first row by $-2$.

	1	-1	-1	1	3
	-2	2	2	-2	-6

Step 2: Add the Result to the Second Row

Now, we will add the result from Step 1 to the second row.

	1	-1	-1	1	3
	-2	2	2	-2	-6
	0	1	-3	-6	0
	2	0	-1	5	12
	7	4	3	6	5

Performing the addition, we get:

	1	-1	-1	1	3
	-2	2	2	-2	-6
	0	3	-1	-8	-6
	2	0	-1	5	12
	7	4	3	6	5

Step 3: Write the New Matrix

The new matrix after performing the operation $-2R_1 + R_2$ is:

{ \left[ \begin{array}{rrrr|r} 1 & -1 & -1 & 1 & 3 \\ 0 & 3 & -1 & -8 & -6 \\ 2 & 0 & -1 & 5 & 12 \\ 7 & 4 & 3 & 6 & 5 \end{array} \right] \}

Conclusion

In this article, we performed the matrix row operation $-2R_1 + R_2$ on a given starting matrix. We multiplied the first row by $-2$ and added the result to the second row, resulting in a new matrix. This operation is a fundamental tool in linear algebra and is used extensively in solving systems of linear equations and manipulating matrices.

Discussion

Matrix row operations are a crucial tool in linear algebra, and understanding how to perform these operations is essential for solving systems of linear equations and manipulating matrices. In this article, we performed a specific matrix row operation and wrote the new matrix. This operation can be used to solve systems of linear equations, find the inverse of a matrix, and perform other matrix manipulations.

Future Work

In future work, we can explore other matrix row operations, such as adding or subtracting multiples of one row from another row, or multiplying a row by a scalar. We can also use matrix row operations to solve systems of linear equations and manipulate matrices in various ways.

References

• [1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
• [2] Matrix Algebra, 2nd Edition, by James E. Gentle

Glossary

• Matrix row operation: An operation that involves adding or subtracting multiples of one row from another row, or multiplying a row by a scalar.
• Linear equation: An equation in which the variables are raised to the power of 1.
• System of linear equations: A set of linear equations that are solved simultaneously.
• Matrix: A rectangular array of numbers or variables.
• Inverse of a matrix: A matrix that, when multiplied by the original matrix, results in the identity matrix.
  Matrix Row Operations: A Q&A Guide =====================================

Introduction

In our previous article, we performed a matrix row operation and wrote the new matrix. In this article, we will answer some frequently asked questions about matrix row operations.

Q: What is a matrix row operation?

A: A matrix row operation is an operation that involves adding or subtracting multiples of one row from another row, or multiplying a row by a scalar.

Q: Why are matrix row operations important?

A: Matrix row operations are important because they can be used to solve systems of linear equations, find the inverse of a matrix, and perform other matrix manipulations.

Q: What are some common matrix row operations?

A: Some common matrix row operations include:

• Adding or subtracting multiples of one row from another row
• Multiplying a row by a scalar
• Interchanging two rows
• Multiplying a row by a non-zero scalar and adding the result to another row

Q: How do I perform a matrix row operation?

A: To perform a matrix row operation, you need to follow these steps:

1. Identify the rows that you want to operate on.
2. Determine the operation that you want to perform (e.g. add, subtract, multiply).
3. Perform the operation on the rows.
4. Write the new matrix.

Q: What are some examples of matrix row operations?

A: Here are some examples of matrix row operations:

• Adding 2 times the first row to the second row:

{ \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 5 & 6 & 7 & 8 \end{array} \right] \}

becomes

{ \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 2 & 3 & 6 & 10 \\ 5 & 6 & 7 & 8 \end{array} \right] \}

• Subtracting 3 times the second row from the first row:

{ \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 5 & 6 & 7 & 8 \end{array} \right] \}

becomes

{ \left[ \begin{array}{rrr|r} 1 & -1 & -3 & -5 \\ 0 & 1 & 2 & 3 \\ 5 & 6 & 7 & 8 \end{array} \right] \}

Q: Can I use matrix row operations to solve systems of linear equations?

A: Yes, you can use matrix row operations to solve systems of linear equations. By performing row operations on the augmented matrix, you can transform it into row-echelon form, which makes it easier to solve the system.

Q: What is row-echelon form?

A: Row-echelon form is a form of a matrix in which all the entries below the leading entry of each row are zero. This form is useful for solving systems of linear equations.

Q: How do I transform a matrix into row-echelon form?

A: To transform a matrix into row-echelon form, you need to perform row operations on the matrix until it is in row-echelon form. This can be done using the following steps:

1. Perform row operations to get a leading entry in the first row.
2. Perform row operations to get zeros below the leading entry in the first row.
3. Perform row operations to get a leading entry in the second row.
4. Perform row operations to get zeros below the leading entry in the second row.
5. Continue this process until the matrix is in row-echelon form.

Conclusion

In this article, we answered some frequently asked questions about matrix row operations. We discussed what matrix row operations are, why they are important, and how to perform them. We also provided some examples of matrix row operations and discussed how to transform a matrix into row-echelon form.