I am studying a definition of a determinant in a textbook.   The left
the index of summation out, which makes it very confusing.  I am going
to type the definition they wrote here.  I am hoping someone could
tell me what the index should be.  Usually there is something like i=1
underneath the big sigma symbol, and then an n above the symbol (which
would tell you to sum the terms one by one up to the number n).  But
they did put any index of summation in the definition.  Here it is:

"det(A) = Ó (+/-)  (a1j1)(a2j2)...(anjn) where the summation is over
all permutations j1j2...jn of the set S={1,2,....n}.  The sign is
taken as + or - according to whether the permutation j1j2...jn is even
or odd."

Is the index of summation supposed to be j=1 to n?  I can't understand
why the left this information off.

So j is not a number: it's a function. They didn't leave the
information off.

For example, in a 3x3 matrix, you will look at the set S={1,2,3}.
There are six permutaitons (functions from S to S that are bijective);
let's call them I, r, r^2, x, y and z. They are:

I:   1|->1, 2|->2, 3|->3.
r:  1|->2, 2|->3, 3|->1.
r^2 : r|->3, 2|->1, 3|->2.
x: 1|->1, 2|->3, 3|->2.
y: 1|->3, 2|->2, 3|->1.
z: 1|->2, 2|->1, 3|->3.

The set of all permutations is then {I, r, r^2, x, y, z}. The sign of
these permutaions is: x, y, and z are odd permutations, I, r, and r^2
are even permutations.

So the sum is taken over all j in {I, r, r^2, x, y, z}. When j=I, you
get the term a_{11}a_{22}a_{33}. When j=r, you get the term a_{12}a_
{23}a_{31} (the second index is the image of the first under the
permutation you are looking at). When j=r^2, you get a_{13}a_{21}a_
{32}. When j=x you get (-1)a_{11}a_{23}a_{32}. When j=y you get (-1)a_
{13}a_{22}a_{31}. When j=z, you get (-1)a_{12}a_{21}a_{33}.

So the summation, taken over all j in {I, r, r^2, x, y, z}, will
yield:

det(A) = a_{11}a_{22}a_{33}           [for j=I]
              + a_{12}a_{23}a_{31}       [for j=r]
              + a_{13}a_{21}a_{32}       [for j=r^2]
              - a_{11}a_{23}a_{32}        [for j=x]
              - a_{13}a_{22}a_{31}        [for j=y]
              - a_{12}a_{21}a_{33}        [for j=z].

det(A) = A11.A22 - A12.A21

det(A) = A11.A22.A33 + A12.A23.A31 + A13.A21.A32 - A11.A23.A32 - A12.A21.A33 - A13.A22.A31

In the matrix elements Aij there is the second index j.

In the summation there is not a single index j, but rather a multi-index [j1, j2, ... jn].
This multi-index is spread out, so to say, over all factors of each product in the
determinant formula. It consists of the second indexes of the factors.
It takes as its values all permutations of 1...n.

Now turning to the 3rd-order formula:

the multi-index takes the values 1,2,3; 2,3,1; 3,1,2 and 1,3,2; 2,1,3; 3,2,1.
the first three are the even permutations of 1,2,3; the remaining three are the odd
permutations.

Please also read http://en.wikipedia.org/wiki/Determinant - you will find there exactly
the definition of your textbook. The j_i is there represented by sigma(i).

Take note of the rule of Sarrus, which refers specifically to 3rd-order determinants.

  sum_{s in S_n} sgn(s) product_{i=1}^n a_{i,s(i)}