Is it possible to generate non-invertible rectangular matrices? If so, how we can prove it? And is there any way to generate randomly such matrices in MATLAB (with preferably uniform distribution)?
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A wide (resp. tall) matrix $A$ has no right (resp. left) inverses if and only if it has deficient row (resp. column) rank, i.e. if and only if the rows (resp. columns) of $A$ are linearly dependent. Owing to the dependency between rows/columns, what you mean by "uniform distribution" is unclear.
At any rate, here are some methods that you may consider. Without loss of generality, suppose $A$ is an $m\times n$ wide matrix (so that $m<n$). Presumably, the entries of $A$ are real numbers.