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...Existence of the determinant learning goals students learn that really exists and find some formulas for it so far our formula is product pivots this isn t such a good because all we know changing order rows might change or at least sign doesn tell us how depends on any particular entry in matrix need other ways finding not better why practice by reducing multiplying if bother its more theoretically useful than practically theory though give insight let s use properties have found to discover new n start with linearity property says m mn linear each row can split up combination am amn first nn there are terms these be into second until get which possible combinations one element from times but many indeed most zero matrices ones same column will two identical thus only remaining where different permutations permutation recognize as p used reduction swap formulize little now definition function whose domain image both set numbers has k either after around changes becomes i sgn correspon...