- 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.__...
- 2.6.20E: 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. b) A.c) A V A V A.
- 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
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.
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.
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 edge-node 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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)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.
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).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , 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.