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To determine if the system \( A\mathbf{x} = \mathbf{b} \) is consistent, we must apply a fundamental concept from linear algebra regarding the rank of matrices.

### Concept:
For a system of linear equations \( A\mathbf{x} = \mathbf{b} \) to be consistent, the rank of the matrix \( A \) should be equal to the rank of the augmented matrix \( [A | \mathbf{b}] \). Mathematically, this can be expressed as:

\[
\text{rank}(A) = \text{rank}([A | \mathbf{b}])
\]

#### Explanation:
- \( A \) is an \( m \times n \) matrix.
- \( \mathbf{x} \) is a vector in \( \mathbb{R}^n \).
- \( \mathbf{b} \) is a vector in \( \mathbb{R}^m \).
- The matrix \( [A | \mathbf{b}] \) represents the augmented matrix obtained by appending \( \mathbf{b} \) as an additional column to \( A \).

If \( \text{rank}(A) \neq \text{rank}([A | \mathbf{b}]) \), the system is inconsistent, meaning no solution exists. However, if \( \text{rank}(A) = \text{rank}([A | \mathbf{b}]) \), the system is consistent, implying that there is at least one solution.

### Answer:
Based on this principle, the correct option is:

**d. \( \text{rank}(A) = \text{rank}([A | \mathbf{b}]) \)**

Would you like more details or further clarification?

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Here are some related questions you might be interested in:

1. How does the rank condition relate to the number of solutions of a system of equations?
2. What happens if \( \text{rank}(A) = \text{rank}([A | \mathbf{b}]) \) but is less than the number of columns in \( A \)?
3. Can you provide an example where \( \text{rank}(A) \neq \text{rank}([A | \mathbf{b}]) \) and explain why it's inconsistent?
4. How is the rank of a matrix defined mathematically?
5. What does it mean geometrically when the system \( A\mathbf{x} = \mathbf{b} \) is consistent?

**Tip**: Always remember that consistency in a system of linear equations hinges on comparing the ranks of the coefficient matrix and its augmented counterpart.

Let A be a matrix and x, b be vectors. The system Ax = b is consistent if...

This problem explores the rank condition for the consistency of a linear system Ax = b, where A is a matrix and b is a vector. Learn the key concepts of linear algebra related to matrix rank and augmented matrices.

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