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The lengths of matrix incidence algebras are studied when their radicals have square zero. All realizable values of the length function are provided for such algebras. In order to obtain this result, a discrete optimization problem is posed and solved. Also, the exact formula of the length is deduced under the additional assumption that the algebra is maximal by inclusion. Moreover, the solution to the length realizability problem is established for matrix incidence algebras with arbitrary radicals under a restriction on the cardinality of the ground field.

On the lengths of matrix incidence algebras with radicals of square zero