Page:Elementary algebra (1896).djvu/455

546. Minors. From the preceding article,

Also from Art. 544,

We shall now explain a simple method of writing down the expansion of a determinant of the third order, and it should be noticed that it is immaterial whether we develop it from the first row or the first column.

From equation (1) we see that the coefficient of any one of the constituents $a_{1},a_{2},a_{3}$, is that determinant of the second order which is obtained by omitting the row and column in which it occurs. These determinants are called the minors of the original determinant, and the left-hand side of equation (1) may be written

$a_{1}A_{1}-a_{2}A_{2}+a_{3}A_{3}$,

where $A_{1},A_{2},A_{3}$, are the minors of $a_{1},a_{2},a_{3}$ respectively. Again, from equation (2), the determinant is equal to

$a_{1}A_{1}-b_{1}B_{1}+c_{1}C_{1}$,

where $A_{1},B_{1},C_{1}$ are the minors of $a_{1},b_{1},c_{1}$ respectively.

547. The determinant

hence