Linear Algebra

310 Chapter 4. Determinants
We finish with a summary (although the final subsection contains the unfinished
business of proving the two theorems). Determinant functions exist,
are unique, and we know how to compute them. As for what determinants are
about, perhaps these lines [Kemp] help make it memorable.
Determinant none,
Solution: lots or none.
Determinant some,
Solution: just one.
Exercises
These summarize the notation used in this book for the 2- and3-permutations. i 1 2 i 1 2 3
φ1(i) 1 2 φ1(i) 1 2 3
φ2(i) 2 1 φ2(i) 1 3 2
φ3(i) 2 1 3
φ4(i) 2 3 1
φ5(i) 3 1 2
φ6(i) 3 2 1
� 3.14 Compute
�
the
�
determinant
�
by using
�
the permutation expansion.
�1
2 3�
� 2 2 1�
� � � �
(a) �4
5 6�
(b) � 3 −1 0�
�
7 8 9
� �
−2 0 5
�
� 3.15 Compute these both with Gauss’ method and with the permutation expansion
formula.
� � � �
�
(a) �2
1�
�0
1 4�
� � �
�3 1�
(b) �0
2 3�
�
1 5 1
�
� 3.16 Use the permutation expansion formula to derive the formula for 3×3 determinants.
3.17 List all of the 4-permutations.
3.18 A permutation, regarded as a function from the set {1, .., n} to itself, is oneto-one
and onto. Therefore, each permutation has an inverse.
(a) Find the inverse of each 2-permutation.
(b) Find the inverse of each 3-permutation.
3.19 Prove that f is multilinear if and only if for all �v, �w ∈ V and k1,k2 ∈ R, this
holds.
f(�ρ1,...,k1�v1 + k2�v2,...,�ρn) =k1f(�ρ1,...,�v1,...,�ρn)+k2f(�ρ1,...,�v2,...,�ρn)
3.20 Find the only nonzero term in the permutation expansion of this matrix.
� �
�0
1 0 0�
� �
�1
0 1 0�
� �
�0
1 0 1�
�0
0 1 0�
Compute that determinant by finding the signum of the associated permutation.
3.21 How would determinants change if we changed property (4) of the definition
to read that |I| =2?
3.22 Verify the second and third statements in Corollary 3.13.