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?

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.