 9.3.12E: a. How many ways can the letters of the word THEORY be arranged in ...
 9.3.1E: a. How many bit strings consist of from one through four digits? (S...
 9.3.2E: a. How many strings of hexadecimal digits consist of from one throu...
 9.3.3E: a. How many integers from 1 through 999 do not have any repeated di...
 9.3.4E: How many arrangements in a row of no more than three letters can be...
 9.3.5E: a. How many fivedigit integers (integers from 10,000through 99,999...
 9.3.6E: In a certain state, license plates consist of from zero to three le...
 9.3.7E: In another state, all license plates consist of from four to six sy...
 9.3.8E: At a certain company, passwords must be from 3–5 symbols long and c...
 9.3.9E: a. Consider the following algorithm segment: How many times will th...
 9.3.10E: A calculator has an eightdigit display and a decimal point that is...
 9.3.11E: a. How many ways can the letters of the word QUICK be arranged in a...
 9.3.13E: A group of eight people are attending the movies together.a. Two of...
 9.3.14E: An early compiler recognized variable names according to the follow...
 9.3.15E: Identifiers in a certain database language must begin with a letter...
 9.3.16E: a. If any seven digits could be used to form a telephone number, ho...
 9.3.17E: a. How many strings of four hexadecimal digits do not have any repe...
 9.3.18E: Just as the difference rule gives rise to a formula for the probabi...
 9.3.19E: A combination lock requires three selections of numbers, each from ...
 9.3.20E: a. How many integers from 1 through 100,000 contain the digit 6 exa...
 9.3.21E: Six new employees, two of whom are married to each other, are to be...
 9.3.22E: Consider strings of length n over the set {a, b, c, d}.a. How many ...
 9.3.23E: a. How many integers from 1 through 1,000 are multiples of 4 or mul...
 9.3.24E: a. How many integers from 1 through 1,000 are multiples of 2 or mul...
 9.3.25E: Counting Strings:a. Make a list of all bit strings of lengths zero,...
 9.3.26E: Counting Strings: Consider the set of all strings of a’s, b’s, and ...
 9.3.27E: For each integer n ? 0, let ak be the number of bit strings of leng...
 9.3.28E: For each integer n ? 2 let an be the number of permutations of {1, ...
 9.3.29E: Refer to Example 9.3.5.a. Write the following IP address in dotted ...
 9.3.30E: A row in a classroom has n seats. Let sn be the number of ways none...
 9.3.31E: Assume that birthdays are equally likely to occur in any one of the...
 9.3.32E: Assuming that all years have 365 days and all birthdays occur with ...
 9.3.33E: A college conducted a survey to explore the academic interests and ...
 9.3.34E: A study was done to determine the efficacy of three different drugs...
 9.3.35E: An interesting use of the inclusion/exclusion rule is to check surv...
 9.3.36E: Fill in the reasons for each step below. If A and B are sets in a f...
 9.3.37E: For each of exercises below, the number of elements in a certain se...
 9.3.38E: For each of exercises below, the number of elements in a certain se...
 9.3.39E: For each of exercises below, the number of elements in a certain se...
 9.3.40E: For use the definition of the Euler phi functionEuler phi functionF...
 9.3.41E: For use the definition of the Euler phi functionEuler phi functionF...
 9.3.42E: A gambler decides to play successive games of blackjack until he lo...
 9.3.43E: A derangement of the set {1, 2,…, n} is a permutation that moves ev...
 9.3.44E: Note that a product x1 x2x3 may be parenthesized in two different w...
 9.3.45E: Use mathematical induction to prove Theorem.TheoremThe Addition Rul...
 9.3.46E: Prove the inclusion/exclusion rule for two sets A and B by showing ...
 9.3.47E: Prove the inclusion/exclusion rule for three sets.
 9.3.48E: Use mathematical induction to prove the general inclusion/exclusion...
 9.3.49E: A circular disk is cut into n distinct sectors, each shaped like a ...
Solutions for Chapter 9.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 9.3
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Since 49 problems in chapter 9.3 have been answered, more than 28060 students have viewed full stepbystep solutions from this chapter. Chapter 9.3 includes 49 full stepbystep solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Solvable system Ax = b.
The right side b is in the column space of A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.