The incorrect statement can be identified by analyzing each of the provided statements with respect to the properties of linear algebra and matrices:

Key Points to Consider:

1. Rank of a matrix: Determines the number of independent rows or columns.
2. System of equations:
   • If $R(A) = R(K) < n$: Consistent with infinitely many solutions.
   • If $R(A) = R(K) = n$: Consistent with a unique solution.
   • If $R(A) \neq R(K)$: Inconsistent with no solution.
   • The rank $R(A)$ cannot exceed $n$ for a coefficient matrix of size $n \times n$.

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Analyzing the statements:

1. If $R(A) = R(K) < n$:
   The system is consistent and has infinitely many solutions.
   ✅ This is correct.

2. If $R(A) = R(K) = n$:
   The system is consistent and has a unique solution.
   ✅ This is correct.

3. If $R(A) = R(K) > n$:
   This is incorrect because the rank of a coefficient matrix $A$ cannot exceed its dimension $n$ (a contradiction arises).

4. If $R(A) \neq R(K)$:
   The system is inconsistent and has no solution.
   ✅ This is correct.

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Answer: The incorrect statement is

"If $R(A) = R(K) > n$, the equations are consistent and have an infinite number of solutions."

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Would you like a detailed explanation of matrix rank or any of the concepts used here?

Related Questions:

1. What is the geometric meaning of the rank of a matrix?
2. Can you explain the difference between consistent and inconsistent systems of equations?
3. How do augmented matrices help solve systems of equations?
4. How does the rank relate to the number of free variables in a system?
5. Can you provide an example of a system where $R(A) = R(K) < n$?

Tip:

Always ensure the dimensions and properties of matrices are consistent when analyzing systems of equations. Misinterpreting rank can lead to incorrect conclusions.