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(b)

Solution (a)

Since each elementary product has two factors, and since each factor comes from a different row, an elementary product can be
written in the form

where the blanks designate column numbers. Since no two factors in the product come from the same column, the column numbers

must be or . Thus the only elementary products are and .

Solution (b)

Since each elementary product has three factors, each of which comes from a different row, an elementary product can be written
in the form

Since no two factors in the product come from the same column, the column numbers have no repetitions; consequently, they must

form a permutation of the set {1, 2, 3}. These permutations yield the following list of elementary products.

As this example points out, an matrix A has elementary products. They are the products of the form , where

is a permutation of the set . By a signed elementary product from A we shall mean an elementary

product multiplied by +1 or . We use the + if is an even permutation and the if

is an odd permutation.

EXAMPLE 6 Signed Elementary Products
List all signed elementary products from the matrices

(a)

(b)

Solution

(a)

Elementary Product Associated Permutation Even or Odd Signed Elementary Product