To determine the correct description of the system of equations represented by the matrix, let's analyze it step by step:

1. Matrix Analysis:

   • The given augmented matrix represents a linear system of equations: 1 & 6 & -4 & | & 4 \\ 0 & 1 & 4 & | & -4 \\ 0 & 0 & 0 & | & 6 \end{bmatrix}$$
   • The third row translates to $0 = 6$, which is a contradiction.

2. Interpretation:

   • The third row implies that the system of equations is inconsistent, as no solutions can satisfy this row. This contradiction means the system cannot have any solution.

3. Conclusion:

   • The system has no solution.

Thus, the correct choice is D. The system has no solution.

Would you like a detailed explanation of how this contradiction is determined?

Here are some related questions to consider:

1. What happens if there is no contradiction in the last row of a matrix?
2. How do we identify when a system has infinitely many solutions?
3. What does the rank of a matrix tell us about its solutions?
4. How can augmented matrices be used to determine system consistency?
5. What is the importance of reduced row echelon form in solving systems?

Tip: Always check for contradictions in augmented matrices when analyzing the solution set.