3     -1      2
-6     2      4
3       1     2
Find rank of matrix

To find the rank of a matrix, we need to reduce it to its row echelon form (or reduced row echelon form) and count the number of non-zero rows. Let's work through the matrix:

\[
\begin{pmatrix}
3 & -1 & 2 \\
-6 & 2 & 4 \\
3 & 1 & 2
\end{pmatrix}
\]

### Step 1: Apply Row Operations
We start by eliminating the entries below the first pivot, which is 3 in the first row.

1. **Row 2:** To eliminate the first element of the second row, add 2 times the first row to the second row:
   \[
   R_2 \to R_2 + 2R_1
   \]
   \[
   \begin{pmatrix}
   3 & -1 & 2 \\
   0 & 0 & 8 \\
   3 & 1 & 2
   \end{pmatrix}
   \]

2. **Row 3:** To eliminate the first element of the third row, subtract the first row from the third row:
   \[
   R_3 \to R_3 - R_1
   \]
   \[
   \begin{pmatrix}
   3 & -1 & 2 \\
   0 & 0 & 8 \\
   0 & 2 & 0
   \end{pmatrix}
   \]

### Step 2: Further Simplify
Now, we focus on the third row. The second pivot is 2 in the third row. To simplify further:

1. **Row 3:** Divide the third row by 2 to get a simpler form:
   \[
   R_3 \to \frac{1}{2}R_3
   \]
   \[
   \begin{pmatrix}
   3 & -1 & 2 \\
   0 & 0 & 8 \\
   0 & 1 & 0
   \end{pmatrix}
   \]

2. **Row 2:** Eliminate the last element in row 2 by subtracting 8 times row 3 from row 2:
   \[
   R_2 \to R_2 - 8R_3
   \]
   \[
   \begin{pmatrix}
   3 & -1 & 2 \\
   0 & 0 & 0 \\
   0 & 1 & 0
   \end{pmatrix}
   \]

### Step 3: Count Non-Zero Rows
The matrix is now in row echelon form. The non-zero rows are:
\[
\begin{pmatrix}
3 & -1 & 2 \\
0 & 1 & 0
\end{pmatrix}
\]
There are **2** non-zero rows, so the rank of the matrix is **2**.

---

Do you need further details or have any questions?

Here are 5 related questions you can explore:
1. What is the significance of a matrix's rank?
2. How is the rank of a matrix connected to its determinant?
3. What happens when the rank of a matrix equals the number of columns?
4. How can the nullity of a matrix be found using its rank?
5. Can the rank of a matrix be greater than the number of rows?

**Tip:** Always ensure that the matrix is simplified correctly to row echelon form when calculating its rank.

Find the rank of the matrix: 3 -1 2; -6 2 4; 3 1 2

Learn how to find the rank of a 3x3 matrix using row operations and row echelon form. This problem demonstrates matrix rank concepts, pivot elements, and matrix row reduction steps. Suitable for advanced high school and undergraduate students studying linear algebra.

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