Find the rank of the matrix
[ 1 2 3]
[1 4 2]
[2 6 5]

To find the rank of a matrix, we perform row reduction (Gaussian elimination) to convert it to its row echelon form (REF). The rank is the number of non-zero rows in the row echelon form.

The given matrix is:

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

### Step 1: Subtract the first row from the second row
\[
R_2 = R_2 - R_1
\]
\[
\begin{bmatrix}
1 & 2 & 3 \\
0 & 2 & -1 \\
2 & 6 & 5
\end{bmatrix}
\]

### Step 2: Subtract 2 times the first row from the third row
\[
R_3 = R_3 - 2R_1
\]
\[
\begin{bmatrix}
1 & 2 & 3 \\
0 & 2 & -1 \\
0 & 2 & -1
\end{bmatrix}
\]

### Step 3: Subtract the second row from the third row
\[
R_3 = R_3 - R_2
\]
\[
\begin{bmatrix}
1 & 2 & 3 \\
0 & 2 & -1 \\
0 & 0 & 0
\end{bmatrix}
\]

The matrix is now in row echelon form. There are 2 non-zero rows, so the rank of the matrix is:

\[
\text{Rank} = 2
\]

Would you like further details or have any questions?

Here are 5 questions that expand on this topic:
1. What are the differences between row echelon form and reduced row echelon form (RREF)?
2. How is the rank related to the solutions of a system of linear equations?
3. Can the rank of a matrix be greater than its number of rows or columns?
4. What is the connection between the determinant of a matrix and its rank?
5. How do you determine the nullity of a matrix using its rank?

**Tip:** The rank of a matrix is always equal to the number of leading entries (pivots) in its row echelon form.

Find the rank of the matrix [1 2 3] [1 4 2] [2 6 5]

Learn how to find the rank of the matrix [1 2 3] [1 4 2] [2 6 5] using Gaussian elimination. This detailed solution walks through each row operation step and explains how to determine the matrix's rank by converting it to row echelon form (REF). Suitable for students in Grades 10-12 or college-level linear algebra courses.

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