This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 368 8. MODULES for a variable matrix over ZŒfxi;j g. This in turn implied the identity for
arbitrary matrices over an arbitrary commutative ring with identity. Exercises 8.3
8.3.1. Show that the set of multilinear maps is an abelian group under
8.3.2. Show that if ' W M n ! N is multilinear, and 2 Sn , then '
is also multilinear. Show that each of the following sets is invariant under
the action of Sn : the symmetric multilinear functions, the skew–symmetric
multilinear functions, and the alternating multilinear functions.
(c) Show that .Rn /k has no nonzero alternating multilinear functions with values in R, if k > n.
Show that .Rn /k has nonzero alternating multilinear functions
with values in R, if k Ä n.
Conclude that Rn is not isomorphic to Rm as R–modules, if m ¤
n. 8.3.4. Compute the following determinant by row reduction. Observe that
the result is an integer, even though the computations involve rational numbers.
det 44 3 15
8.3.5. Prove the cofactor expansion identity
det.A/ D n
X . 1/i Cj ai;j det.Ai;j /: j D1 by showing that the right hand side deﬁnes an alternating multilinear function of the columns of the matrix A whose value at the identity matrix is 1.
It follows from Corollary 8.3.8 that the right hand is equal to the determinant of A
8.3.6. Prove a cofactor expansion by columns: For ﬁxed j ,
det.A/ D n
. 1/i Cj ai;j det.Ai;j /:
i D1 8.3.7. Prove Cramer’s rule: If A is an invertible n–by–n matrix over R,
and b 2 Rn , then the unique solution to the matrix equation Ax D b is
xj D det.A/ 1 det.Aj /; ...
View Full Document