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Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
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Invertibility of one matrix constructed by order n subgroup of symmetric group
The answer is no. Consider the subgroup generated by (1243). The matrix you have will be cyclic. See Determinant of cyclic matrix, proof without eigenvectors for a discussion of the determinant.
It is …