Math Problem Statement
Find rank
Solution
It seems that your image shows a matrix and steps involved in reducing it through row operations to find its rank.
Let's summarize the process of finding the rank of a matrix:
- Forming the matrix: The rank of a matrix is the number of linearly independent rows or columns in it.
- Row-reducing the matrix: Using elementary row operations (swapping rows, multiplying rows by a non-zero scalar, and adding multiples of one row to another), you reduce the matrix to its row echelon form or reduced row echelon form.
- Count the non-zero rows: The rank of the matrix is the number of non-zero rows after it has been reduced.
Looking at the steps outlined in your image:
- The matrix has been row-reduced through some elementary row operations.
- Based on the final matrix in the image, the non-zero rows appear to be clearly marked.
I'll interpret and solve this further for rank computation:
- Original matrix: 1 & 3 & -1 & 3 \\ 2 & 4 & 3 & -2 \\ 3 & 6 & 0 & 1 \end{pmatrix}$$
- Elementary row operations: It seems that rows are being manipulated, as indicated in the notations , etc.
Once the matrix is row-reduced, count the number of non-zero rows. Based on the reduced matrix:
1 & 3 & -1 & 3 \\ 0 & 0 & 4 & -3 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ From this, there are **2 non-zero rows**. ### **Conclusion:** The **rank of the matrix is 2**, since there are two non-zero rows after row reduction. Would you like a detailed breakdown of the row operations used in this process? Here are five relative questions based on matrix rank: 1. What are elementary row operations, and how do they affect the rank of a matrix? 2. How do you find the rank of a matrix using Gaussian elimination? 3. Can the rank of a matrix be determined using its columns instead of its rows? 4. How does the rank of a matrix relate to the solutions of a system of linear equations? 5. What is the difference between row echelon form and reduced row echelon form? **Tip**: The rank of a matrix can never exceed the smaller of its row or column count.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Reduction
Formulas
Rank of a matrix is the number of non-zero rows in its row echelon form
Theorems
Rank-Nullity Theorem
Gaussian Elimination
Suitable Grade Level
University level, Linear Algebra (1st or 2nd year)
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