 2.1: In Exercises 16, perform the indicated matrix operations.
 2.2: In Exercises 16, perform the indicated matrix operations.
 2.3: In Exercises 16, perform the indicated matrix operations.
 2.4: In Exercises 16, perform the indicated matrix operations.
 2.5: In Exercises 16, perform the indicated matrix operations.
 2.6: In Exercises 16, perform the indicated matrix operations.
 2.7: In Exercises 710, write out the system of linear equations represen...
 2.8: In Exercises 710, write out the system of linear equations represen...
 2.9: In Exercises 710, write out the system of linear equations represen...
 2.10: In Exercises 710, write out the system of linear equations represen...
 2.11: In Exercises 1114, write the system of linear equations in matrix f...
 2.12: In Exercises 1114, write the system of linear equations in matrix f...
 2.13: In Exercises 1114, write the system of linear equations in matrix f...
 2.14: In Exercises 1114, write the system of linear equations in matrix f...
 2.15: In Exercises 1518, find and
 2.16: In Exercises 1518, find and
 2.17: In Exercises 1518, find and
 2.18: In Exercises 1518, find and
 2.19: In Exercises 1922, find the inverse of the matrix (if it exists).
 2.20: In Exercises 1922, find the inverse of the matrix (if it exists).
 2.21: In Exercises 1922, find the inverse of the matrix (if it exists).
 2.22: In Exercises 1922, find the inverse of the matrix (if it exists).
 2.23: In Exercises 23 26, write the system of linear equations in the for...
 2.24: In Exercises 23 26, write the system of linear equations in the for...
 2.25: In Exercises 23 26, write the system of linear equations in the for...
 2.26: In Exercises 23 26, write the system of linear equations in the for...
 2.27: In Exercises 27 and 28, find A
 2.28: In Exercises 27 and 28, find A
 2.29: In Exercises 29 and 30, find such that the matrix is nonsingular.
 2.30: In Exercises 29 and 30, find such that the matrix is nonsingular.
 2.31: In Exercises 31 and 32, find the inverse of the elementary matrix.
 2.32: In Exercises 31 and 32, find the inverse of the elementary matrix.
 2.33: In Exercises 3336, factor into a product of elementary matrices.
 2.34: In Exercises 3336, factor into a product of elementary matrices.
 2.35: In Exercises 3336, factor into a product of elementary matrices.
 2.36: In Exercises 3336, factor into a product of elementary matrices.
 2.37: Find two matrices such that
 2.38: Find two matrices such that
 2.39: Find three matrices such that
 2.40: Find matrices and such that
 2.41: (a) Find scalars and such that (b) Show that there do not exist sca...
 2.42: Show that if then
 2.43: Let and be nonsingular matrices. Prove that is nonsingular by showi...
 2.44: Writing Let and be matrices and let be nonsingular. If is it true t...
 2.45: In Exercises 45 and 46, find the LUfactorization of the matrix.
 2.46: In Exercises 45 and 46, find the LUfactorization of the matrix.
 2.47: In Exercises 47 and 48, use the LUfactorization of the coefficient...
 2.48: In Exercises 47 and 48, use the LUfactorization of the coefficient...
 2.49: (a) Addition of matrices is not commutative. (b) The transpose of t...
 2.50: (a) The product of a matrix and a matrix is a matrix that is (b) Th...
 2.51: (a) All matrices are invertible. (b) If an matrix is not symmetric,...
 2.52: (a) If and are matrices and is invertible, then (b) If and are nons...
 2.53: At a convenience store, the numbers of gallons of 87 octane, 89 oct...
 2.54: At a certain dairy mart, the numbers of gallons of skim, 2%, and wh...
 2.55: The numbers of calories burned by individuals of different body wei...
 2.56: The final grades in a particular linear algebra course at a liberal...
 2.57: In Exercises 57 and 58, determine whether the matrix is stochastic.
 2.58: In Exercises 57 and 58, determine whether the matrix is stochastic.
 2.59: In Exercises 59 and 60, use the given matrix of transition probabil...
 2.60: In Exercises 59 and 60, use the given matrix of transition probabil...
 2.61: A country is divided into 3 regions. Each year, 10% of the resident...
 2.62: Find the steady state matrix for the populations described in Exerc...
 2.63: In Exercises 63 and 64, find the uncoded row matrices of the indica...
 2.64: In Exercises 63 and 64, find the uncoded row matrices of the indica...
 2.65: In Exercises 6568, find to decode the cryptogram. Then decode the m...
 2.66: In Exercises 6568, find to decode the cryptogram. Then decode the m...
 2.67: In Exercises 6568, find to decode the cryptogram. Then decode the m...
 2.68: In Exercises 6568, find to decode the cryptogram. Then decode the m...
 2.69: In Exercises 69 and 70, use a graphing utility or computer software...
 2.70: In Exercises 69 and 70, use a graphing utility or computer software...
 2.71: An industrial system has two industries with the input requirements...
 2.72: An industrial system with three industries has the inputoutput mat...
 2.73: In Exercises 7376, find the least squares regression line for the p...
 2.74: In Exercises 7376, find the least squares regression line for the p...
 2.75: In Exercises 7376, find the least squares regression line for the p...
 2.76: In Exercises 7376, find the least squares regression line for the p...
 2.77: A farmer used four test plots to determine the relationship between...
 2.78: The Consumer Price Index (CPI) for all items for the years 2001 to ...
 2.79: The table shows the average monthly cable television rates in the U...
 2.80: The table shows the numbers of cellular phone subscribers (in milli...
 2.81: The table shows the average salaries (in millions of dollars) of Ma...
Solutions for Chapter 2: Matrices
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 2: Matrices
Get Full SolutionsSince 81 problems in chapter 2: Matrices have been answered, more than 18117 students have viewed full stepbystep solutions from this chapter. Chapter 2: Matrices includes 81 full stepbystep solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.