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To find the rank of the matrix \( A \), which is given as:

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

we will perform row reduction to bring the matrix to its row echelon form, and then count the number of non-zero rows. Let's compute this step-by-step.

---

### Step 1: Write the matrix
\[
A = \begin{bmatrix} 
1 & 2 & 3 \\ 
1 & 4 & 2 \\ 
2 & 6 & 5 
\end{bmatrix}.
\]

---

### Step 2: Eliminate entries below the first pivot (row 1, column 1)
Subtract row 1 from row 2:
\[
R_2 = R_2 - R_1 \implies \begin{bmatrix} 
1 & 2 & 3 \\ 
0 & 2 & -1 \\ 
2 & 6 & 5 
\end{bmatrix}.
\]

Next, eliminate row 3, column 1, by subtracting \( 2 \times R_1 \) from \( R_3 \):
\[
R_3 = R_3 - 2R_1 \implies \begin{bmatrix} 
1 & 2 & 3 \\ 
0 & 2 & -1 \\ 
0 & 2 & -1 
\end{bmatrix}.
\]

---

### Step 3: Eliminate entries below the pivot in column 2
Subtract \( R_2 \) from \( R_3 \):
\[
R_3 = R_3 - R_2 \implies \begin{bmatrix} 
1 & 2 & 3 \\ 
0 & 2 & -1 \\ 
0 & 0 & 0 
\end{bmatrix}.
\]

---

### Step 4: Count the non-zero rows
The matrix in row-echelon form is:
\[
\begin{bmatrix} 
1 & 2 & 3 \\ 
0 & 2 & -1 \\ 
0 & 0 & 0 
\end{bmatrix}.
\]

There are **2 non-zero rows**, so the rank of the matrix is:
\[
\text{Rank}(A) = 2.
\]

Let me know if you'd like further clarification or additional examples!

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### Related Questions:
1. How is rank related to the determinant of a square matrix?
2. What is the significance of a full-rank matrix?
3. How would you find the rank of a \( 4 \times 4 \) matrix?
4. How does row echelon form differ from reduced row echelon form?
5. Can a non-square matrix have a rank equal to its largest dimension?

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### Tip:
When finding the rank of a matrix, always focus on row operations that simplify the matrix systematically to identify the number of linearly independent rows.

Learn how to calculate the rank of a matrix through row reduction to row echelon form. This step-by-step solution demonstrates key concepts in linear algebra, including elementary row operations and the significance of matrix rank. Suitable for high school and early college students.

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