Dante Is Solving The System Of Equations Below:${ \begin{array}{l} r + S - T = -4 \ 2r - 3s + T = 1 \ 3r - 2s + 2t = 3 \end{array} }$He Writes The Row Echelon Form Of The Matrix. Which Matrix Did Dante Write?A.
Introduction
In mathematics, solving systems of equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. One of the methods used to solve systems of equations is the row echelon form (REF) of a matrix. In this article, we will explore how to write the row echelon form of a matrix and apply it to solve a system of equations.
What is Row Echelon Form?
Row echelon form is a method of transforming a matrix into a simpler form by performing elementary row operations. The resulting matrix has the following properties:
- All the entries below the leading entry (the first non-zero entry in each row) are zero.
- The leading entry in each row is to the right of the leading entry in the row above it.
- The leading entry in each row is a 1.
Step 1: Write the Augmented Matrix
To write the row echelon form of a matrix, we first need to write the augmented matrix of the system of equations. The augmented matrix is a matrix that combines the coefficients of the variables with the constant terms.
For the system of equations given by Dante:
{ \begin{array}{l} r + s - t = -4 \\ 2r - 3s + t = 1 \\ 3r - 2s + 2t = 3 \end{array} \}
The augmented matrix is:
{ \begin{array}{ccc|c} 1 & 1 & -1 & -4 \\ 2 & -3 & 1 & 1 \\ 3 & -2 & 2 & 3 \end{array} \}
Step 2: Perform Elementary Row Operations
To write the row echelon form of the matrix, we need to perform elementary row operations. These operations involve multiplying a row by a non-zero constant, adding a multiple of one row to another row, or interchanging two rows.
Let's perform the following elementary row operations:
- Multiply the first row by 2 and add it to the second row.
- Multiply the first row by 3 and add it to the third row.
The resulting matrix is:
{ \begin{array}{ccc|c} 1 & 1 & -1 & -4 \\ 0 & -5 & 3 & 9 \\ 0 & -5 & 5 & 15 \end{array} \}
Step 3: Write the Row Echelon Form
Now that we have performed the elementary row operations, we can write the row echelon form of the matrix.
The row echelon form of the matrix is:
{ \begin{array}{ccc|c} 1 & 1 & -1 & -4 \\ 0 & 1 & -\frac{3}{5} & -\frac{9}{5} \\ 0 & 0 & 1 & 3 \end{array} \}
Conclusion
In this article, we have explored how to write the row echelon form of a matrix and apply it to solve a system of equations. We have seen that the row echelon form of a matrix has the following properties:
- All the entries below the leading entry are zero.
- The leading entry in each row is to the right of the leading entry in the row above it.
- The leading entry in each row is a 1.
We have also seen how to perform elementary row operations to write the row echelon form of a matrix. By following these steps, we can solve systems of equations using the row echelon form of a matrix.
Final Answer
The final answer is:
{
\begin{array}{ccc|c}
1 & 1 & -1 & -4 \\
0 & 1 & -\frac{3}{5} & -\frac{9}{5} \\
0 & 0 & 1 & 3
\end{array}
\}$<br/>
**Frequently Asked Questions about Row Echelon Form**
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A: The purpose of row echelon form is to simplify a matrix by performing elementary row operations, making it easier to solve systems of equations. A: A matrix is in row echelon form if it satisfies the following conditions: A: Elementary row operations are a set of operations that can be performed on a matrix to transform it into a simpler form. These operations include: A: To perform elementary row operations, follow these steps: A: Row echelon form and reduced row echelon form are two different forms of a matrix. The main difference between them is that in reduced row echelon form, all the entries above the leading entry are also zero. A: To convert a matrix from row echelon form to reduced row echelon form, follow these steps: A: Yes, you can use row echelon form to solve systems of equations with more than three variables. However, the process may be more complex and require more steps. A: Yes, there are limitations to using row echelon form. For example: A: Yes, you can use row echelon form to solve systems of equations with complex coefficients. However, the process may be more complex and require more steps. A: Yes, there are other methods for solving systems of equations besides row echelon form. Some of these methods include: In this article, we have answered some of the most frequently asked questions about row echelon form. We have covered topics such as the purpose of row echelon form, how to know if a matrix is in row echelon form, and how to perform elementary row operations. We have also discussed the limitations of using row echelon form and other methods for solving systems of equations.Q: What is the purpose of row echelon form?
Q: How do I know if a matrix is in row echelon form?
Q: What are elementary row operations?
Q: How do I perform elementary row operations?
Q: What is the difference between row echelon form and reduced row echelon form?
Q: How do I convert a matrix from row echelon form to reduced row echelon form?
Q: Can I use row echelon form to solve systems of equations with more than three variables?
Q: Are there any limitations to using row echelon form?
Q: Can I use row echelon form to solve systems of equations with complex coefficients?
Q: Are there any other methods for solving systems of equations besides row echelon form?
Conclusion