To determine the incorrect statement, let us analyze each one carefully based on the principles of linear algebra.

Key Points:

1. Rank of Coefficient Matrix $A$: $R(A)$ represents the rank of the coefficient matrix.
2. Rank of Augmented Matrix $K$: $R(K)$ represents the rank of the augmented matrix (formed by appending the column of constants to $A$).
3. Consistency: A system of equations is consistent if $R(A) = R(K)$. Otherwise, it is inconsistent.
4. Number of Solutions:
   • $R(A) = R(K) < n$: Consistent with infinite solutions.
   • $R(A) = R(K) = n$: Consistent with a unique solution.
   • $R(A) ≠ R(K)$: Inconsistent, so no solutions.

Analysis of the Statements:

1. $R(A) = R(K) < n$: Correct. The system is consistent and has an infinite number of solutions because the rank is less than $n$, meaning free variables exist.

2. $R(A) = R(K) = n$: Correct. The system is consistent and has a unique solution because the rank equals the number of variables $n$, meaning there are no free variables.

3. $R(A) = R(K) > n$: Incorrect. The rank of a matrix cannot exceed its number of columns (in this case $n$). Therefore, this case is not possible.

4. $R(A) ≠ R(K)$: Correct. The system is inconsistent and has no solutions if the ranks are different.

Incorrect Statement:

• $R(A) = R(K) > n$:

Would you like further details or a deeper explanation on ranks, consistency, and solutions?

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Related Questions:

1. What is the significance of rank in determining the solutions of a system of linear equations?
2. How do free variables influence the number of solutions in a consistent system?
3. Can a matrix have a rank greater than the number of its rows or columns? Why or why not?
4. What is the geometric interpretation of the rank of a matrix in terms of linear independence?
5. How is the augmented matrix used to solve systems of linear equations?

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Tip: To quickly check if a system is consistent, compare the ranks of $A$ and $K$. If they differ, the system is inconsistent.