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To determine the rank of the given matrix:

\[
\begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & 2 & 2 \end{bmatrix}
\]

we need to find the maximum number of linearly independent rows or columns. Here’s the process:

1. **Start with Row Reduction:** Transform the matrix into row echelon form.
   \[
   \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & 2 & 2 \end{bmatrix}
   \]
   
2. **Make Row Operations:**
   - Subtract 2 times Row 1 from Row 3:
     \[
     \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 2 \end{bmatrix}
     \]
   - Divide Row 3 by 2:
     \[
     \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}
     \]
   - Subtract Row 2 from Row 3 to make it unique:
     \[
     \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}
     \]

3. **Count Non-zero Rows:** There are 2 non-zero rows left, which makes the rank of this matrix **2**.

Thus, the answer is:

**The rank of the matrix is 2.**

Would you like a more detailed explanation on any step? Here are some related questions:

1. How do you determine the rank of a matrix in general?
2. What is the significance of rank in a matrix?
3. How does row echelon form help in finding the rank?
4. Why is rank important in solving linear systems?
5. Can rank be determined by looking at the columns instead of rows?

**Tip:** The rank of a matrix is the maximum number of linearly independent rows or columns, which corresponds to the number of pivots in its row echelon form.

The rank of the matrix \[ \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & 2 & 2 \end{bmatrix} \] is:

Learn how to determine the rank of the matrix \[ \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & 2 & 2 \end{bmatrix} \] by using row reduction to find the maximum number of linearly independent rows. This problem covers essential concepts in linear algebra suitable for college-level studies.

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