Why Matrix Row Operations are Important

Oct 13, 2025 by windailys.com 36 views

Question 1: Perform the following matrix row operation and write the new matrix

Original Matrix

The given matrix is a 3x4 matrix, which can be represented as:

{ \begin{bmatrix} 5 & 10 & -15 & -40 \\ -7 & 3 & 2 & 0 \\ 2 & 0 & 7 & 5 \end{bmatrix} \}

Matrix Row Operation

To perform a matrix row operation, we need to select a row and multiply it by a scalar, add a multiple of another row to it, or interchange two rows. In this case, let's perform the following operation:

Row Operation: Multiply the first row by 2 and add 3 times the second row to it.

Step 1: Multiply the first row by 2

To multiply the first row by 2, we multiply each element of the first row by 2:

{ \begin{bmatrix} 10 & 20 & -30 & -80 \\ -7 & 3 & 2 & 0 \\ 2 & 0 & 7 & 5 \end{bmatrix} \}

Step 2: Add 3 times the second row to the first row

To add 3 times the second row to the first row, we multiply each element of the second row by 3 and add it to the corresponding element of the first row:

{ \begin{bmatrix} 10 & 20 & -30 & -80 \\ -7 & 3 & 2 & 0 \\ 2 & 0 & 7 & 5 \end{bmatrix} \}

{ \begin{bmatrix} 10 + (-21) & 20 + 9 & -30 + 6 & -80 + 0 \\ -7 & 3 & 2 & 0 \\ 2 & 0 & 7 & 5 \end{bmatrix} \}

{ \begin{bmatrix} -11 & 29 & -24 & -80 \\ -7 & 3 & 2 & 0 \\ 2 & 0 & 7 & 5 \end{bmatrix} \}

New Matrix

The new matrix after performing the row operation is:

{ \begin{bmatrix} -11 & 29 & -24 & -80 \\ -7 & 3 & 2 & 0 \\ 2 & 0 & 7 & 5 \end{bmatrix} \}

Discussion Category: Mathematics

Matrix row operations are an essential tool in linear algebra, and they can be used to solve systems of linear equations. In this lesson, we performed a matrix row operation to transform the original matrix into a new matrix. This new matrix can be used to solve a system of linear equations.

Why Matrix Row Operations are Important

Matrix row operations are important because they can be used to:

Solve systems of linear equations: Matrix row operations can be used to transform a matrix into a form that can be used to solve a system of linear equations.
Find the inverse of a matrix: Matrix row operations can be used to find the inverse of a matrix, which is an essential tool in linear algebra.
Solve systems of linear inequalities: Matrix row operations can be used to solve systems of linear inequalities, which is an essential tool in optimization.

Conclusion

In this lesson, we performed a matrix row operation to transform the original matrix into a new matrix. This new matrix can be used to solve a system of linear equations. Matrix row operations are an essential tool in linear algebra, and they can be used to solve systems of linear equations, find the inverse of a matrix, and solve systems of linear inequalities.

Key Takeaways

Matrix row operations: Matrix row operations are an essential tool in linear algebra, and they can be used to solve systems of linear equations, find the inverse of a matrix, and solve systems of linear inequalities.
Solving systems of linear equations: Matrix row operations can be used to solve systems of linear equations.
Finding the inverse of a matrix: Matrix row operations can be used to find the inverse of a matrix.

Practice Problems

Perform the following matrix row operation and write the new matrix:

{ \begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 & 2 & 3 & 4 \\ 3 & 6 & 9 & 12 \end{bmatrix} \}

Operation: Multiply the first row by 2 and add 3 times the second row to it.

Perform the following matrix row operation and write the new matrix:

{ \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 3 & 6 & 9 & 12 \end{bmatrix} \}

Operation: Multiply the second row by 2 and add 3 times the third row to it.

Answer Key

$\begin{bmatrix} 4 & 8 & 12 & 16 \\ 1 & 2 & 3 & 4 \\ 3 & 6 & 9 & 12 \end{bmatrix} \}$$ 2. $${ \begin{bmatrix} 1 & 2 & 3 & 4 \\ 8 & 16 & 24 & 32 \\ 3 & 6 & 9 & 12 \end{bmatrix} \}$$<br/> **Lesson 38: Solve ${$3 \times 3\$}$ Systems**$

Q&A: Matrix Row Operations

Q: What is a matrix row operation?

A: A matrix row operation is a mathematical operation that involves multiplying a row of a matrix by a scalar, adding a multiple of another row to it, or interchanging two rows.

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 solve systems of linear inequalities.

Q: How do I perform a matrix row operation?

A: To perform a matrix row operation, you need to select a row and multiply it by a scalar, add a multiple of another row to it, or interchange two rows.

Q: What are some common matrix row operations?

A: Some common matrix row operations include:

Multiplying a row by a scalar: This involves multiplying each element of a row by a scalar.
Adding a multiple of another row to it: This involves adding a multiple of another row to the selected row.
Interchanging two rows: This involves swapping two rows.

Q: How do I use matrix row operations to solve a system of linear equations?

A: To use matrix row operations to solve a system of linear equations, you need to transform the matrix into a form that can be used to solve the system. This can involve multiplying a row by a scalar, adding a multiple of another row to it, or interchanging two rows.

Q: What are some tips for performing matrix row operations?

A: Some tips for performing matrix row operations include:

Be careful when multiplying a row by a scalar: Make sure to multiply each element of the row by the scalar.
Be careful when adding a multiple of another row to it: Make sure to add the correct multiple of the other row.
Be careful when interchanging two rows: Make sure to swap the correct rows.

Q: How do I know when to use matrix row operations?

A: You should use matrix row operations when you need to solve a system of linear equations, find the inverse of a matrix, or solve a system of linear inequalities.

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

A: Some common mistakes to avoid when performing matrix row operations include:

Multiplying a row by the wrong scalar: Make sure to multiply each element of the row by the correct scalar.
Adding the wrong multiple of another row to it: Make sure to add the correct multiple of the other row.
Interchanging the wrong rows: Make sure to swap the correct rows.

Q: How do I check my work when performing matrix row operations?

A: To check your work when performing matrix row operations, you should:

Verify that the row operations are correct: Make sure that the row operations are performed correctly.
Verify that the resulting matrix is correct: Make sure that the resulting matrix is correct.

Conclusion

Matrix row operations are an essential tool in linear algebra, and they can be used to solve systems of linear equations, find the inverse of a matrix, and solve systems of linear inequalities. By understanding how to perform matrix row operations, you can solve a wide range of problems in linear algebra.