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.