Select The Correct Answer.What Is The Row Echelon Form Of This Matrix?$\[ \begin{bmatrix} 1 & 7 & 8 \\ -1 & 3 & 2 \\ 1 & 5 & 0 \end{bmatrix} \\]A. \[$\begin{bmatrix} 1 & 7 & 8 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\$\]B.
Introduction
In linear algebra, a matrix can be transformed into its row echelon form (REF) through a series of elementary row operations. The row echelon form is a fundamental concept in solving systems of linear equations and finding the inverse of a matrix. In this article, we will explore the row echelon form of a given matrix and provide a step-by-step guide on how to transform it into its REF.
What is Row Echelon Form?
The row echelon form of a matrix is a matrix that has the following properties:
- All the entries below the leading entry (the first non-zero entry in a row) are zero.
- Each leading entry is to the right of the leading entry in the row above it.
- The leading entry in each row is a 1.
Given Matrix
The given matrix is:
{ \begin{bmatrix} 1 & 7 & 8 \\ -1 & 3 & 2 \\ 1 & 5 & 0 \end{bmatrix} \}
Step 1: Apply Elementary Row Operations
To transform the given matrix into its row echelon form, we need to apply a series of elementary row operations. The first step is to eliminate the -1 in the second row by adding the first row to the second row.
{ \begin{bmatrix} 1 & 7 & 8 \\ 0 & 10 & 10 \\ 1 & 5 & 0 \end{bmatrix} \}
Step 2: Eliminate the 1 in the Third Row
Next, we need to eliminate the 1 in the third row by subtracting the first row from the third row.
{ \begin{bmatrix} 1 & 7 & 8 \\ 0 & 10 & 10 \\ 0 & -2 & -8 \end{bmatrix} \}
Step 3: Make the Leading Entry in the Second Row a 1
Now, we need to make the leading entry in the second row a 1 by dividing the second row by 10.
{ \begin{bmatrix} 1 & 7 & 8 \\ 0 & 1 & 1 \\ 0 & -2 & -8 \end{bmatrix} \}
Step 4: Eliminate the -2 in the Third Row
Finally, we need to eliminate the -2 in the third row by adding 2 times the second row to the third row.
{ \begin{bmatrix} 1 & 7 & 8 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} \}
Conclusion
The row echelon form of the given matrix is:
{ \begin{bmatrix} 1 & 7 & 8 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} \}
This is the correct answer.
Discussion
The row echelon form of a matrix is a fundamental concept in linear algebra. It is used to solve systems of linear equations and find the inverse of a matrix. In this article, we have provided a step-by-step guide on how to transform a matrix into its row echelon form. We have also discussed the properties of the row echelon form and how to apply elementary row operations to achieve it.
References
- [1] Strang, G. (1988). Linear Algebra and Its Applications. 3rd ed. San Diego: Harcourt Brace Jovanovich.
- [2] Lay, D. C. (2005). Linear Algebra and Its Applications. 3rd ed. Boston: Addison-Wesley.
Frequently Asked Questions
- Q: What is the row echelon form of a matrix? A: The row echelon form of a matrix is a matrix that has the following properties: all the entries below the leading entry are zero, each leading entry is to the right of the leading entry in the row above it, and the leading entry in each row is a 1.
- Q: How do I transform a matrix into its row echelon form? A: To transform a matrix into its row echelon form, you need to apply a series of elementary row operations. The first step is to eliminate the -1 in the second row by adding the first row to the second row.
- Q: What are the properties of the row echelon form?
A: The properties of the row echelon form are: all the entries below the leading entry are zero, each leading entry is to the right of the leading entry in the row above it, and the leading entry in each row is a 1.
Row Echelon Form Q&A: Frequently Asked Questions =====================================================
Introduction
In our previous article, we discussed the row echelon form of a matrix and provided a step-by-step guide on how to transform a matrix into its row echelon form. In this article, we will answer some frequently asked questions about the row echelon form.
Q: What is the row echelon form of a matrix?
A: The row echelon form of a matrix is a matrix that has the following properties: all the entries below the leading entry are zero, each leading entry is to the right of the leading entry in the row above it, and the leading entry in each row is a 1.
Q: How do I transform a matrix into its row echelon form?
A: To transform a matrix into its row echelon form, you need to apply a series of elementary row operations. The first step is to eliminate the -1 in the second row by adding the first row to the second row.
Q: What are the properties of the row echelon form?
A: The properties of the row echelon form are: all the entries below the leading entry are zero, each leading entry is to the right of the leading entry in the row above it, and the leading entry in each row is a 1.
Q: Can a matrix have multiple row echelon forms?
A: No, a matrix can only have one row echelon form. The row echelon form is a unique form of a matrix.
Q: How do I find the inverse of a matrix using the row echelon form?
A: To find the inverse of a matrix using the row echelon form, you need to follow these steps:
- Transform the matrix into its row echelon form.
- Swap the rows of the matrix to get a 1 in the top-left corner.
- Multiply the first row by the reciprocal of the leading entry.
- Subtract the first row from the other rows to get zeros below the leading entry.
- Repeat steps 2-4 for each row.
Q: Can a matrix have a row echelon form with zeros on the diagonal?
A: Yes, a matrix can have a row echelon form with zeros on the diagonal. This is known as a singular matrix.
Q: How do I determine if a matrix is singular or not?
A: To determine if a matrix is singular or not, you need to check if the determinant of the matrix is zero. If the determinant is zero, then the matrix is singular.
Q: What is the relationship between the row echelon form and the reduced row echelon form?
A: The row echelon form and the reduced row echelon form are related but distinct concepts. The reduced row echelon form is a matrix that has the same properties as the row echelon form, but with the additional property that all the entries above the leading entry are zero.
Conclusion
In this article, we have answered some frequently asked questions about the row echelon form of a matrix. We have discussed the properties of the row echelon form, how to transform a matrix into its row echelon form, and how to find the inverse of a matrix using the row echelon form.
References
- [1] Strang, G. (1988). Linear Algebra and Its Applications. 3rd ed. San Diego: Harcourt Brace Jovanovich.
- [2] Lay, D. C. (2005). Linear Algebra and Its Applications. 3rd ed. Boston: Addison-Wesley.
Frequently Asked Questions
- Q: What is the row echelon form of a matrix? A: The row echelon form of a matrix is a matrix that has the following properties: all the entries below the leading entry are zero, each leading entry is to the right of the leading entry in the row above it, and the leading entry in each row is a 1.
- Q: How do I transform a matrix into its row echelon form? A: To transform a matrix into its row echelon form, you need to apply a series of elementary row operations. The first step is to eliminate the -1 in the second row by adding the first row to the second row.
- Q: What are the properties of the row echelon form? A: The properties of the row echelon form are: all the entries below the leading entry are zero, each leading entry is to the right of the leading entry in the row above it, and the leading entry in each row is a 1.