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To determine the rank of the given matrix \( A = \begin{bmatrix} 1 & 2 & -1 \\ -2 & -4 & 2 \\ 3 & 6 & -3 \end{bmatrix} \), we follow the process of reducing the matrix to row echelon form (REF) or reduced row echelon form (RREF) and count the number of non-zero rows. The rank of a matrix is the number of non-zero rows in its row echelon form.

Let me compute the row-reduction steps to determine the rank.The rank of the matrix \( A = \begin{bmatrix} 1 & 2 & -1 \\ -2 & -4 & 2 \\ 3 & 6 & -3 \end{bmatrix} \) is **1**. This means that the rows of this matrix are linearly dependent, and the matrix spans a one-dimensional subspace.

Would you like more details on how the row reduction is done, or any further clarifications?

Here are some related questions you might be interested in:
1. How can we interpret the rank in terms of linear independence?
2. What is the difference between rank and nullity of a matrix?
3. How do we perform Gaussian elimination step-by-step?
4. Can the rank of a matrix change after transposing?
5. How does matrix rank relate to the solution of a system of equations?

**Tip:** When computing the rank of a matrix, remember that it represents the maximum number of linearly independent rows or columns.

Determine the rank of the matrix A = [[1, 2, -1], [-2, -4, 2], [3, 6, -3]].

This problem involves determining the rank of a 3x3 matrix A = [[1, 2, -1], [-2, -4, 2], [3, 6, -3]] using row reduction techniques. Learn how the rank reflects the maximum number of linearly independent rows.

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