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We are given a matrix \( A \) and asked to find the value of \( K \) such that the rank of matrix \( A \), denoted \( \rho(A) \), is 2.

The matrix \( A \) is:

\[
A = \begin{pmatrix}
-1 & 2 & 1 & -2 \\
2 & -4 & -2 & K \\
0 & 1 & -3 & 2
\end{pmatrix}
\]

The rank of a matrix is the number of linearly independent rows or columns in the matrix. Since \( \rho(A) = 2 \), this means that there should be two linearly independent rows in the matrix, and the third row must be a linear combination of the other two rows.

### Step-by-step solution:

1. **Row 1**: \( (-1, 2, 1, -2) \)
2. **Row 2**: \( (2, -4, -2, K) \)
3. **Row 3**: \( (0, 1, -3, 2) \)

We will perform elementary row operations to simplify the matrix and check for conditions that make the rank equal to 2.

### Step 1: Row Reduction

We will start by making Row 2 a multiple of Row 1.

Multiply Row 1 by 2:

\[
\text{Row 1} \times 2 = (-2, 4, 2, -4)
\]

Now add this to Row 2:

\[
\text{Row 2} + \text{Row 1} \times 2 = (2, -4, -2, K) + (-2, 4, 2, -4) = (0, 0, 0, K - 4)
\]

Thus, after this operation, the matrix becomes:

\[
A = \begin{pmatrix}
-1 & 2 & 1 & -2 \\
0 & 0 & 0 & K - 4 \\
0 & 1 & -3 & 2
\end{pmatrix}
\]

### Step 2: Analyze the Rank Condition

- For the rank to be 2, the second row \( (0, 0, 0, K-4) \) should be a zero row, because this would mean that Row 3 must be dependent on the remaining non-zero rows. Therefore, \( K - 4 = 0 \), which gives:

\[
K = 4
\]

Thus, the correct value of \( K \) is \( 4 \), which corresponds to option **D**.

Would you like a detailed breakdown of the steps, or do you have any other questions?

Here are 5 related questions to consider:
1. What is the definition of matrix rank and how is it determined?
2. How do elementary row operations affect the rank of a matrix?
3. What happens when two rows of a matrix are linearly dependent?
4. How can matrix rank help in solving systems of linear equations?
5. How does the concept of rank apply to invertibility of matrices?

**Tip:** When solving matrix rank problems, row reduction (Gaussian elimination) is a very useful technique to simplify the analysis.

If ρ(A) = 2, then K has the value in the matrix A as shown.

Find the value of K such that the matrix A has a rank of 2. This problem involves matrix rank, elementary row operations, and linear dependence. Suitable for college-level students studying linear algebra.

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