 2.6.6E: Repeat Exercise with and .ExerciseConsider the production model x =...
 2.6.9E: Solve the Leontief production equation for an economy with three se...
 2.6.7E: Let C and d be as in Exercise 5a. Determine the production level ne...
 2.6.8E: Let C be an n × n consumption matrix whose column sums are less tha...
 2.6.1E: Exercises 14 refer to an economy that is divided into three sector...
 2.6.2E: Exercises refer to an economy that is divided into three sectorsma...
 2.6.3E: Exercises refer to an economy that is divided into three sectorsma...
 2.6.4E: Exercises refer to an economy that is divided into three sectorsma...
 2.6.5E: a. Compute the transfer matrix of the network in the figure below. ...
 2.6.10E: The consumption matrix C for the U.S economy in 1972 has the proper...
 2.6.11E: The Leontief production equation, x = Cx + d, is usually accompanie...
 2.6.12E: Let C be a consumption matrix such that Find a difference equation ...
 2.6.13E: [M] The consumption matrix C below is based on input–output data fo...
 2.6.14E: [M] The demand vector in Exercise 13 is reasonable for 1958 data, b...
 2.6.15E: [M] Use equation (6) to solve the problem in Exercise 13. Set How m...
Solutions for Chapter 2.6: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 2.6
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 2.6 includes 15 full stepbystep solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 15 problems in chapter 2.6 have been answered, more than 46604 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Outer product uv T
= column times row = rank one matrix.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.