For the matrix  A=[■(1&5&2@4&3&1@6&8&4)] find the rank of the matrix using row reduction method.

To find the rank of the matrix \( A \) using the row reduction method (also known as Gaussian elimination), we need to perform row operations to convert the matrix into its row echelon form (REF). The rank of the matrix will be the number of non-zero rows in its REF.

Given matrix \( A \):

\[
A = \begin{bmatrix} 
1 & 5 & 2 \\ 
4 & 3 & 1 \\ 
6 & 8 & 4 
\end{bmatrix}
\]

### Step 1: Formulate the augmented matrix

We start with the matrix \( A \):

\[
A = \begin{bmatrix} 
1 & 5 & 2 \\ 
4 & 3 & 1 \\ 
6 & 8 & 4 
\end{bmatrix}
\]

### Step 2: Perform elementary row operations

1. **Row 2 transformation:** Make the element in position (2,1) a zero. We subtract 4 times the first row from the second row:
   \[
   R_2 \to R_2 - 4R_1
   \]
   \[
   \begin{bmatrix} 
   1 & 5 & 2 \\ 
   4 - 4(1) & 3 - 4(5) & 1 - 4(2) \\ 
   6 & 8 & 4 
   \end{bmatrix}
   = \begin{bmatrix} 
   1 & 5 & 2 \\ 
   0 & -17 & -7 \\ 
   6 & 8 & 4 
   \end{bmatrix}
   \]

2. **Row 3 transformation:** Make the element in position (3,1) a zero. We subtract 6 times the first row from the third row:
   \[
   R_3 \to R_3 - 6R_1
   \]
   \[
   \begin{bmatrix} 
   1 & 5 & 2 \\ 
   0 & -17 & -7 \\ 
   6 - 6(1) & 8 - 6(5) & 4 - 6(2) 
   \end{bmatrix}
   = \begin{bmatrix} 
   1 & 5 & 2 \\ 
   0 & -17 & -7 \\ 
   0 & -22 & -8 
   \end{bmatrix}
   \]

3. **Row 3 transformation:** Make the element in position (3,2) a zero. We perform the following operation:
   \[
   R_3 \to R_3 - \frac{22}{17} R_2
   \]
   After calculating:
   \[
   \begin{bmatrix} 
   1 & 5 & 2 \\ 
   0 & -17 & -7 \\ 
   0 & 0 & -\frac{10}{17} 
   \end{bmatrix}
   \]

### Step 3: Final row echelon form

The matrix in row echelon form is:

\[
\begin{bmatrix} 
1 & 5 & 2 \\ 
0 & -17 & -7 \\ 
0 & 0 & -\frac{10}{17} 
\end{bmatrix}
\]

### Step 4: Determine the rank

In this row echelon form, all three rows have non-zero elements, so the matrix has 3 non-zero rows. Hence, the **rank** of matrix \( A \) is:

\[
\boxed{3}
\]

---

Would you like more details or any clarifications on the steps? 

Here are five related questions to consider:
1. How do row operations preserve the rank of a matrix?
2. What is the difference between row echelon form (REF) and reduced row echelon form (RREF)?
3. Can the rank of a matrix be greater than the number of rows or columns?
4. How does the rank of a matrix relate to the solutions of a system of linear equations?
5. How would you find the rank of a larger matrix, e.g., a 4x5 matrix?

**Tip:** When reducing matrices, be careful with fractions. It's easier to keep results in fraction form until the very end to avoid rounding errors.

For the matrix A = [1 5 2; 4 3 1; 6 8 4] find the rank of the matrix using row reduction method.

Learn how to find the rank of a matrix A = [1 5 2; 4 3 1; 6 8 4] using the row reduction (Gaussian elimination) method. This problem involves linear algebra concepts, such as matrix operations, row echelon form, and the rank theorem. Suitable for Grades 10-12.

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