8. Any determinant |a , b | formed out of the elements of the original

|a', b'| determinant, by selecting the lines and columns at pleasure, is termed a _minor_ of the original determinant; and when the number of lines and columns, or order of the determinant, is n-1, then such determinant is called a _first minor_; the number of the first minors is = n², the first minors, in fact, corresponding to the several elements of the determinant--that is, the coefficient therein of any term whatever is the corresponding first minor. The first minors, each divided by the determinant itself, form a system of elements _inverse_ to the elements of the determinant. A determinant is _symmetrical_ when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be = 0), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is _skew_; but if the relation does extend to the diagonal terms (that is, if these are each = 0), then the determinant is _skew symmetrical_; thus the determinants |a, h, g|; | a , [nu], - [mu]|; | 0, [nu], - [mu]| |h, b, f| |- [nu], b,[lambda]| |- [nu], 0,[lambda]| |g, f, c| | [mu],-[lambda], c | | [mu],- [lambda], 0| are respectively symmetrical, skew and skew symmetrical: The theory admits of very extensive algebraic developments, and applications in algebraical geometry and other parts of mathematics. For further developments of the theory of determinants see ALGEBRAIC FORMS. (A. CA.)