2. 𝐴 = [1 2 3 2 1 4 3 0 5 ]. Find the Rank of matrix

The given matrix \( A \) is:

\[
A = \begin{bmatrix} 
1 & 2 & 3 \\
2 & 1 & 4 \\
3 & 0 & 5 
\end{bmatrix}
\]

To find the rank of the matrix, we follow these steps:

1. **Form the augmented matrix** and perform row-reducing operations (Gaussian elimination) to get the matrix into its row echelon form (REF).
   
   \[
   A = \begin{bmatrix} 
   1 & 2 & 3 \\
   2 & 1 & 4 \\
   3 & 0 & 5 
   \end{bmatrix}
   \]

   We'll perform the following operations:
   - Subtract 2 times the first row from the second row.
   - Subtract 3 times the first row from the third row.

   After these operations:

   \[
   A = \begin{bmatrix} 
   1 & 2 & 3 \\
   0 & -3 & -2 \\
   0 & -6 & -4 
   \end{bmatrix}
   \]

2. Next, we'll simplify the third row by subtracting 2 times the second row from the third row.

   \[
   A = \begin{bmatrix} 
   1 & 2 & 3 \\
   0 & -3 & -2 \\
   0 & 0 & 0 
   \end{bmatrix}
   \]

3. Now, the matrix is in row echelon form (REF). From this, we can see that there are two non-zero rows. Therefore, the rank of the matrix is **2**.

### Final Answer:
The rank of the matrix \( A \) is **2**.

---

Would you like a detailed breakdown of the steps? Here are five related questions for practice:

1. How do you determine the rank of a matrix using Gaussian elimination?
2. What is the difference between row echelon form and reduced row echelon form?
3. Can a matrix have a rank higher than the number of its rows?
4. How do elementary row operations affect the rank of a matrix?
5. What is the relationship between the rank of a matrix and its determinant?

**Tip:** The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix.

A = [1 2 3 2 1 4 3 0 5]. Find the rank of the matrix.

This problem explains how to find the rank of a 3x3 matrix A = [1 2 3; 2 1 4; 3 0 5] by applying Gaussian elimination and row echelon form. Learn the key steps involved in determining the rank, suitable for undergraduate students studying linear algebra.

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