Linear Algebra. Math-4100, Fall 2007
Assignment 5
Due Thursday, October 18, by 4pm. (Either in class, or my mailbox in AE 301, or
under my door AE 405).
Reading
Oct. 8, 11: Strang Sections 5.1, 5.2.
Oct. 15 and 18: Strang Sections 5.1โ5.3; 7.1; Gelfand Section 9.
Problems
You are encouraged to consult the text and notes and discuss the problems with other people.
However, the solutions should be yours. Please indicated on your papers, who you discussed
the problems with. Please submit extra credit problems on a separate sheet of paper.
1. (a) Problem 5.1 #16.
(b) A skew-symmetric matrix has KT=โK. Use properties of determinants, in
particular linearity and det A= det AT, to prove that all skew symmetric matrices
in odd dimensions are singular.
2. Problem 5.1 #24.
3. Problem 5.1 #25.
4. Problem 5.1 #28.
5. Problem 5.2 #6: Place the smallest number of zeros in a 4 ร4 matrix that will
guarantee det A= 0 no matter how the nonzero entries are selected. Place as many
zeros as possible while still allowing det A6= 0.
6. Problem 5.2 #11: How many permutations of (1,2,3,4) are even and what are they?
A permutation is called even if it can be obtained from (1,2,3,4) with an even number
of exchanges. How many perturbations of (1,2, ..., n) are even?
7. Problem 5.2 #14.
8. Problem 5.2 #25. Note that the key idea for designing counterexamples is contained
in Problem 27.
9. Problem 5.3 #1(b).
10. Problem 5.3 #21.
E5. (Extra credit). Calculate the n-th order determinant
โ =
๎
๎
๎
๎
๎
๎
๎
๎
๎
๎
x a a ... a
a x a ... a
a a x ... a
. . . ... .
a a a ... x
๎
๎
๎
๎
๎
๎
๎
๎
๎
๎
E6. (Based on Problem 5.3 #11 for professors only). If you know all n2cofactors
of a nรninvertible matrix A, how would you find A?