= ab'c" - ab"c' + a'b"c - a'bc" + a"bc' - a"b'c.
3. It further appears that a determinant is a linear function[1] of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs. It at once follows that, if two columns are identical, or if two lines are identical, the value of the determinant is = 0. It may be added, that if the lines are converted into columns, and the columns into lines, in such a way as to leave the dexter diagonal unaltered, the value of the determinant is unaltered; the determinant is in this case said to be _transposed_.
4. By what precedes it appears that there exists a function of the n² elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient +1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations. Observe that the properties show at once that if any column is = 0 (that is, if the elements in the column are each = 0), then the determinant is = 0; and further, that if any two columns are identical, then the determinant is = 0.
5. Reverting to the system of linear equations written down at the beginning of this article, consider the determinant
|ax + by + cz - d , b , c |; |a'x + b'y + c'z - d', b', c'| |a"x + b"y + c"z - d", b", c"|