order determinants to fit in with the above pattern. The simplest way to define these
higher order determinants is in terms of second order determinants. Thus, we define a
3 ×3 determinant by:
∣
∣
∣
∣
∣a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣
∣
∣
∣
∣=a 11∣
∣
∣
∣a 22 a 23
a 32 a 33∣
∣
∣
∣−a^12∣
∣
∣
∣a 21 a 23
a 31 a 33∣
∣
∣
∣+a^13∣
∣
∣
∣a 21 a 22
a 31 a 32∣
∣
∣
∣=a 11 a 22 a 33 −a 11 a 23 a 32 −a 12 a 21 a 33 +a 12 a 23 a 31
+a 13 a 21 a 32 −a 13 a 22 a 31The elements of the first row are multiplied by the determinant obtained by crossing out
the row and column of the element, with an alternating sign. This is called theexpansion
of the determinant by the first row. In fact, one can expand by any row or column and
we now show how to do this for general order determinants. Let
n=∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
a 11 a 12 ... a 1 n
a 21 a 22 ... a 2 n
..
...
...
.
an 1 an 2 ... ann∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣be a generalnth order determinant. Choose any elementaij.Thecofactorofaij, denoted
Aij,isthe(n−1)th order determinant(− 1 )i+jDijwhereDijis the determinant obtained
by deleting theirow andjcolumn ofn(Dijis called theminorofaij).
Then the generalnth order determinant can be expressed in terms of (n−1)th order
determinants (and hence in terms of (n−2)th order, etc., down to second order) byan
expansion in the cofactors of any row or column. For example, for expansion ofnby
the first row we have
n=a 11 A 11 +a 12 A 12 +a 13 A 13 +...+a 1 nA 1 nTake the 3×3 case as an example:
3 =∣
∣
∣
∣
∣a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣
∣
∣
∣
∣The cofactors of the second row are
A 21 =(− 1 )^2 +^1∣
∣
∣
∣a 12 a 13
a 32 a 33∣
∣
∣
∣A 22 =(− 1 )^2 +^2∣
∣
∣
∣a 11 a 13
a 31 a 33∣
∣
∣
∣