- 1.9.1: An automobile mechanic (M) and a body shop (B) use each other's ser...
- 1.9.2: A simple economy produces food (F) and housing (H). The production ...
- 1.9.3: Consider the open economy described by the accompanying table, wher...
- 1.9.4: company produces Web design, software, and networking services. Vie...
- 1.9.5: In Exercises 56, use matrix inversion to find the production vector...
- 1.9.6: In Exercises 56, use matrix inversion to find the production vector...
- 1.9.7: Consider an open economy with consumption matrix (a) Showthat the e...
- 1.9.8: Consider an open economy with consumption matrix If the open sector...
- 1.9.9: Consider an open economy with consumption matrix Show that the Leon...
- 1.9.10: (a) Consider an open economy with a consumption matrix C whose colu...
- 1.9.11: Prove: If C is an matrix whose entries are nonnegative and whose ro...
- 1.9.a: In parts (a)(e) determine whether the statement is true or false, a...
- 1.9.b: In parts (a)(e) determine whether the statement is true or false, a...
- 1.9.c: In parts (a)(e) determine whether the statement is true or false, a...
- 1.9.d: In parts (a)(e) determine whether the statement is true or false, a...
- 1.9.e: In parts (a)(e) determine whether the statement is true or false, a...
Solutions for Chapter 1.9: Leontief Input-Output Models
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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).
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
A directed graph that has constants Cl, ... , Cm associated with the edges.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. 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.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).