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Determine all possibilities for the solution set of a homogeneous system of 5 equations in 4 unknowns and the rank of the system is 3.

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Determine all possibilities for the solution set of a homogeneous system of 5 equations in 4 unknowns and the rank of the system is 3.

(a) finite

(b) zero or finitely many

(c) only one

(d) infinite

This question was posed to me during an online exam.

This is a very interesting question from Counting topic in portion Counting of Discrete Mathematics

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Right option is (d) infinite

Best explanation: A homogeneous system is consistent. The rank is r = 3 and the number of variables is n = 4. Hence there is n – r = 1 free variable. Thus there are infinitely many solutions.

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