Homogeneous and nonhomogeneous matrices basics

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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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2 Answers

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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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