# Homogeneous and nonhomogeneous matrices basics

If $A$ is an $m\times n$ matrix where $m\lt n$

The nonhomogeneous system $Ax=o$ has at least one solution and the homogeneous system $Ax=0$ has a unique solution.

Are the above statements true or false ...please assist

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Do you have two separate statements? – David Mitra Feb 19 '12 at 16:09
The first system looks pretty homogeneous. – Dylan Moreland Feb 19 '12 at 16:23
@Dylan That may be a bad edit of mine. Reverting it... – David Mitra Feb 19 '12 at 16:39
You will have to take more care, Bulelwa, when you post questions here. Your first sentence ends in mid-air. Your non-homogeneous system isn't non-homogeneous. Your spelling leaves much to be desired. Voting to close. – Gerry Myerson Feb 20 '12 at 4:49
@Gerry: The first question ended in "mid-air" because of the use of <; of course, the OP should look at how the question looks after posting. Th first system as Ax=o, as opposed to the second which is Ax=0; however, o is a rather poor choice for "something different from 0". – Arturo Magidin Feb 20 '12 at 5:01

If $m\lt n$ then the system has fewer equations ($m$) than unknowns ($n$) (variables) and it is not possible for the homogeneous system to have a unique solution. The nonhomogeneous system may not have a solution.

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True, the matrix has more unknowns than rows than unknowns, so there must be free variables, which means that there must be several solutions for the non-homogeneous system, but only one for the homogeneous system.

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