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Let's calculate the ranks of the matrices using linear dependence. To find the rank, we need to reduce each matrix to its row echelon form (or reduced row echelon form) and count the number of non-zero rows.

I'll start by analyzing each matrix one by one.

### Matrix a)
\[
\begin{pmatrix}
1 & -2 & -1 & 4 \\
-2 & 4 & 2 & -8 \\
4 & -8 & -4 & 16
\end{pmatrix}
\]
We will reduce this matrix by using Gaussian elimination.

### Matrix b)
\[
\begin{pmatrix}
1 & -2 & -2 & 4 \\
0 & 4 & 0 & -8 \\
-2 & -8 & -4 & 16
\end{pmatrix}
\]

### Matrix c)
\[
\begin{pmatrix}
1 & -2 & -1 & 4 \\
-2 & 4 & 2 & -8 \\
4 & -8 & -4 & 16
\end{pmatrix}
\]

### Matrix d)
\[
\begin{pmatrix}
3 & -2 & 1 & 6 \\
-2 & 0 & 2 & 0 \\
1 & -2 & 3 & 6
\end{pmatrix}
\]

I'll perform the Gaussian elimination and compute the ranks.The ranks of the matrices are as follows:

- **Matrix a:** Rank = 1
- **Matrix b:** Rank = 3
- **Matrix c:** Rank = 1
- **Matrix d:** Rank = 2

Let me know if you would like further details on how each matrix was reduced or if you have any questions. Here are some related problems you can explore:

1. How is the rank of a matrix related to the number of pivot columns?
2. What is the relationship between matrix rank and the determinant?
3. How does row echelon form help in finding the rank?
4. How would the rank change if an additional row or column was added?
5. What is the geometric interpretation of matrix rank in terms of linear independence?

**Tip:** The rank of a matrix is equal to the number of linearly independent rows or columns it contains!

Calcula los rangos de las siguientes matrices utilizando la dependencia lineal.

Learn how to calculate the rank of matrices using Gaussian elimination and linear dependence principles. This problem involves matrices and explores linear dependence to determine the rank. Suitable for students studying Linear Algebra at the college level.

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