Grassmann Algebra

TheExteriorProduct.nb 27
Α2 � � � Αn � a11 ������ e1 ����� � a12 ������ e2 ����� � � � a1�n ������� en �����
If we now premultiply by the first factor Α1 we get a particularly symmetric form.
Α1 � Α2 � � � Αn � �a11�e1 � a12�e2 � � � a1�n�en�
� �a11 ������ e1 ����� � a12 ������ e2 ����� � � � a1�n ������� en ����� �
Multiplying this out and remembering that the exterior product of a basis element with the
cobasis element of another basis element is zero, we get the sum:
Α1 � Α2 � � � Αn � �a11�a11 ������ � a12�a12 ������ � � � a1�n�a1�n ������� ��e1 � e2 � � � en
But we have already seen that:
Α1 � Α2 � � � Αn � De1 � e2 � � � en
Hence the determinant D can be expressed as the sum of products:
D � a11�a11 ������ � a12�a12 ������ � � � a1�n�a1�n �������
showing that the aij
������ are the cofactors of the aij .
Mnemonically we can visualize the aij as the determinant with the row and column containing
������
aij as 'struck out' by the underscore.
Of course, there is nothing special in the choice of the element Αi about which to expand the
determinant. We could have written the expansion in terms of any factor (row or column):
D � ai1�ai1 ������ � ai2�ai2 ������ � � � ain�ain ������
It can be seen immediately that the product of the elements of a row with the cofactors of any
other row is zero, since in the exterior product formulation the row must have been included in
the calculation of the cofactors.
Summarizing these results for both row and column expansions we have:
n
�
j�1
aij�akj
������
n
� �
j�1
aji�ajk � D ∆ik
������
In matrix terms of course this is equivalent to the standard result A Ac T � DI or equivalently
A�1 Ac
T
� �������
D , where Ac is the matrix of cofactors of the elements of A.
The Laplace expansion in cofactor form
We have already introduced the Laplace expansion technique in our discussion of determinants:
1. Take the exterior product of any m of the Αi
2. Take the exterior product of the remaining nÐm of the Αi
3. Take the exterior product of the results of the first two operations (in the correct order).
In this section we revisit it more specifically in the context of the cofactor. The results will be
important in deriving later results.
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