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To find the rank of matrix \( A \), we need to analyze its rows and identify any linear dependencies.

The given matrix \( A \) is:

\[
A = \begin{pmatrix} 1 & 3 & -4 & -2 \\ -2 & -6 & 8 & 4 \\ 7 & 21 & -28 & -14 \end{pmatrix}
\]

Let's go through the steps to determine its rank by transforming it to row echelon form or identifying linearly dependent rows.

1. **Row 1**: \( (1, 3, -4, -2) \)
2. **Row 2**: \( (-2, -6, 8, 4) \), which is a scalar multiple of Row 1 (specifically, Row 2 = -2 * Row 1).
3. **Row 3**: \( (7, 21, -28, -14) \), which is also a scalar multiple of Row 1 (Row 3 = 7 * Row 1).

Since Rows 2 and 3 are linear combinations of Row 1, they do not add any new information, and we only have one independent row.

Thus, the rank of the matrix \( A \) is **1**.

Would you like further details or have any questions?

Here are some related questions that may interest you:
1. What is the significance of a matrix’s rank?
2. How do row operations affect the rank of a matrix?
3. Can rank help determine the invertibility of a matrix?
4. How does the rank relate to the solution of a system of linear equations?
5. What is the difference between row rank and column rank?

**Tip:** To determine if rows are linearly dependent, look for scalar multiples or use Gaussian elimination to simplify the matrix.

Learn how to determine the rank of a matrix by identifying linearly dependent rows. This solution involves transforming the matrix to row echelon form and examining each row for dependencies. Ideal for college-level students studying matrix algebra.

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