Rank of an Augmented matrix

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Can the rank of coeffecient matrix be greater than augmented matrix? Also,what is the condition for an inconsistent set of linear equations?

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2 Answers

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The rank of a matrix is the dimension of the span of its columns. The coefficient matrix has fewer columns than the augmented matrix. So, the answer to your first question is no. I don't understand the second one.

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For the system $AX=B$ where $A$ is $n×n$ matrix:

1. $Rank(A)=Rank(C)=n$ implies unique solution.
2. $Rank(A)=Rank(C)<n$ implies infinitely many solutions.
3. $Rank(A)<Rank(C)$ implies no solution (inconsistent).

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