498 EQUATION theorem that a given arrangement can be derived from the primitive arrangement only by an odd number, or else only by an even number of interchanges, a theorem the verifica tion of which may be easily obtained from the theorem (in fact a particular case of the general one), an arrangement can be derived from itself only by an even number of interchanges.] And this being so, each product has the sign belonging to the corresponding arrangement of the columns; in particular, a determinant contains with the sign + the product of the elements in its dexter diagonal. It is to be observed that the rule gives as many positive as negative arrangements, the number of each being = 11.2...?*. The rule of signs may be expressed in a different form. Giving to the columns in the primitive arrangement the numbers 1, 2, 3...?i, to obtain the sign belonging to any other arrangement we take, as often as a lower number succeeds a higher one, the sign -, and, compounding together all these minus signs, obtain the proper sign, + or - as the case may be. Thus, for three columns, it appears by either rule toat 123, 231, 312 are positive; 213, 321, 213 are negative; and the developed expression of the foregoing determinant of the third order is * = 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 inter changed, 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 deter minant 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 fol lows 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 n- 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 = (that is, if the elements in the column are each = 0), then the determinant is = ; 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 deter minant 1 The expression, a linear function, is here used in its narrowest sense, a linear function without constant term; what is meant is, that the determinant is in regard to the elements a, a , a", . . of any column or line thereof, a function of the form Aa 4- A a + A "a" + . . . , without any term independent of a, a , a" ... a x+b y+c z-d a! x + b y + d z - d a"x + b"y + c"z - d b , c , , b" , c" t it appears that this is =* a , b , c + y a! , b , c i a", b", c" b , b , c b , b , c b", b", c" M c ,b , c ,V , c", b", c c c" - d , b , c d ,V ,<1 d", b", c" viz., the second and third terms each vanishing, it is x a , b , c a , b , c a", b", c" d d d ,b,c > ^" , c " But if the linear equations hold good, then the first column of the original determinant is = , and therefore the deter minant itself is = ; that is, the linear equations give b , c b , c b" , c" d , b , c d , b , c d", b", c" = 0; which is the result obtained above. We might in a similar way find the values of y and z, but there is a more symmetrical process. Join to the original equations the new equation ax+Py + yz=S ; a like process shows that, the equations being satisfied, we have a , |8, 7, 8 ; = 0; a , b , c , d a , I , c , d a", b", c", d" or, as this may be written, a , j8 , 7 - 5 a^ , ^ , c | = ; a b c , d a , b , c a b c , d a", b", c" a" b" c", d" which, considering 8 as standing herein for its value ax + fiy + yz , is a consequence of the original equations only : we have thus an expression for ax + fty + yz , an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of a, (3, y on the two sides respectively, we have the values of x,y,z; in fact, these quantities, each multiplied by a , b , c i i a , b , c I a", b", c" i are in the first instance obtained in the forms I a , b , c , d a , b , c , d a", b", c", d" i a , b , c , d a , b , c , d a", b", c", cT< but these are b , c , d I b , c , d | b", c", d d , a I d , a a , b , c , d a , b , c , d a", b", c", d d a , b d a , b c", d" , a" i j d" a", b"{ or, what is the same thing, b , c , d b , c , d b", c", d"[ c , a , d c , a , d c" , a", d" a , b , d a b , d a" b", d" respectively. 6. Multiplication of two determinants of the same order. The theorem is obtained very easily from the last preced ing definition of a determinant. It is most simply ex pressed thus (a , b , c ) (a 1 , b , c } (a", b", e") a , b , c a , b , c a", b", c" where the expression on the left side stands for a determi nant, the terms of the first line being (a, I, c)(a, a , a"), that is, aa + ba + ca", (a, b, c)(J3, ft , ft"), that is, aft + bft + cft",
(a, b, c)(y, y , y"), that is ay + by + cy" ; and similarly the