What does the rank of a matrix tells about the consistency of the given equation?

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- May 4th 2009, 03:49 AMroshanheroRank of a matrix
What does the rank of a matrix tells about the consistency of the given equation?

- May 4th 2009, 05:48 AMShowcase_22
I'm not sure what consistency is when applied to matrices, but here's something I knowabout ranks:

The rank of a matrix is equal to the number of linearly independent equations in the system. - May 4th 2009, 07:15 AMHallsofIvy
A system of equations is "consistent" if there exist at least one solution to the system. If the rank of an n by n coefficient matrix is n, then there exist a unique solution and the system is consistent. If the rank is less than n, then the range of the matrix is a subspace or [itex]R^n[/itex] and there exist a solution if and only if the vector on the right hand side of the equation is in that subspace.