 2.6.1E: ?<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mi>e</...
 2.6.2E: ?Find A + B, where <math xmlns="http://www.w3.org/1998/Math/MathML"...
 2.6.3E: ?Find AB if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a<...
 2.6.4E: ?Find the product AB, where <math xmlns="http://www.w3.org/1998/Mat...
 2.6.5E: Find a matrix A such that [Hint: Finding A requires that you solve ...
 2.6.6E: ?Find a matrix A such that <math xmlns="http://www.w3.org/1998/Math...
 2.6.7E: Let A be an m × n matrix and let 0 be the m × n matrix that has all...
 2.6.8E: Show that matrix addition is commutative; that is, show that if A a...
 2.6.9E: Show that matrix addition is associative: that is, show that if A, ...
 2.6.10E: Let A be a 3 × 4 matrix, B be a 4 × 5 matrix, and C be a 4×4 matrix...
 2.6.11E: What do we know about the sizes of the matrices A and B if both of ...
 2.6.12E: In this exercise we show that matrix multiplication is distributive...
 2.6.13E: In this exercise we show that matrix multiplication is associative....
 2.6.14E: The n × n matrix A =[aij] is called a diagonal matrix i aij = 0 whe...
 2.6.15E: Let Find a formula for An , whenever n is a positive integer.
 2.6.16E: Show that (At)t = A.
 2.6.17E: Let A and B be two n × n matrices. Show thata) (A + B)t = At +Bt.__...
 2.6.18E:
 2.6.19E:
 2.6.20E: Let a) Find A?1. [Hint: Use Exercise 19.]________________b) Find A3...
 2.6.21E: Let A be an invertible matrix. Show that (An)?l = (A?1) n whenever ...
 2.6.22E: Let A be a matrix. Show that the matrix AAt is symmetric. Hint: Sh...
 2.6.23E: Suppose that A is an n × n matrix where n is a positive integer. Sh...
 2.6.24E: ?a) Show That the system of simultaneous linear equations in the va...
 2.6.25E: ?Use Exercises 18 and 24 to solve the system
 2.6.26E: ?Let Find
 2.6.27E: Let Find ________________ ________________
 2.6.28E: ?Find the Boolean product of A and B, where
 2.6.29E: ?Let Finda) A[2]. b) A[3].c) A V A[2] V A[3].
 2.6.30E: Let A be a zero–one matrix. Show thata) A ? A = A. ________________...
 2.6.31E: In this exercise we show that the meet and join operations are comm...
 2.6.32E: In this exercise we show that the meet and join operations are asso...
 2.6.33E: We will establish distributive laws of the meet over the join opera...
 2.6.34E: Let A be an n × n zero–one matrix. Let I be the n × n identity matr...
 2.6.35E: In this exercise we will show that the Boolean product of zero–one ...
Solutions for Chapter 2.6: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 2.6
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Chapter 2.6 includes 35 full stepbystep solutions. Since 35 problems in chapter 2.6 have been answered, more than 437793 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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