https://file.aiquickdraw.com/mathbot/file/1726476376078jfibrqi8.jpg

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:
1. **Forming the matrix**: The rank of a matrix is the number of linearly independent rows or columns in it.
2. **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.
3. **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:

1. **Original matrix**:
   \[
   \begin{pmatrix}
   1 & 3 & -1 & 3 \\
   2 & 4 & 3 & -2 \\
   3 & 6 & 0 & 1
   \end{pmatrix}
   \]

2. **Elementary row operations**:
   It seems that rows are being manipulated, as indicated in the notations \( R_3 - R_1 \), etc.

Once the matrix is row-reduced, count the number of non-zero rows. Based on the reduced matrix:
   \[
   \begin{pmatrix}
   1 & 3 & -1 & 3 \\
   0 & 0 & 4 & -3 \\
   0 & 0 & 0 & 0
   \end{pmatrix}
   \]

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.

This article explains how to find the rank of a matrix using row reduction techniques. The rank is determined by counting the non-zero rows in the matrix's row echelon form after applying elementary row operations. This process applies key concepts from linear algebra, including the Rank-Nullity Theorem and Gaussian Elimination.

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