The (3) condition is trully messy, mean it only considers crossings through i->j arrow, moreover if at least that hold, we should have n-i - (n-j-k) = -i + j + k => k + (-i + j + k) = - i + j + 2k => (-1)^{-i + j + 2k} (-1)^{i + j} Looks like home improvements ...
Thanks very much for this it certainly explained very well the connection between Cofactor expansion and the permutation proofs I have seen online! Still the question for me remains: through my own research out of interest, the determinant of a square matric was presented as the distinguishing feature of invertibility. In other words, if the determinant is non-zero then the reduced-row-echelon-form of the original square matrix satisfies invertibility conditions (pivots in every row and every column etc.) - where please can I find a direct proof that the determinant arrived at by Cofactor expansion in fact implies invertibility?
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The (3) condition is trully messy, mean it only considers crossings through i->j arrow,
moreover if at least that hold, we should have n-i - (n-j-k) = -i + j + k => k + (-i + j + k) = - i + j + 2k => (-1)^{-i + j + 2k} (-1)^{i + j}
Looks like home improvements ...
Amazing! Thank u so much, sir
Thanks very much for this it certainly explained very well the connection between Cofactor expansion and the permutation proofs I have seen online! Still the question for me remains: through my own research out of interest, the determinant of a square matric was presented as the distinguishing feature of invertibility. In other words, if the determinant is non-zero then the reduced-row-echelon-form of the original square matrix satisfies invertibility conditions (pivots in every row and every column etc.) - where please can I find a direct proof that the determinant arrived at by Cofactor expansion in fact implies invertibility?