- 1.6.1E: Suppose an economy has only two sectors: Goods and Services. Each y...
- 1.6.2E: Find another set of equilibrium prices for the economy in Example 1...
- 1.6.3E: Consider an economy with three sectors: Fuels and Power, Manufactur...
- 1.6.4E: Suppose an economy has four sectors: Mining, Lumber, Energy, and Tr...
- 1.6.5E: An economy has four sectors: Agriculture, Manufacturing, Services, ...
- 1.6.6E: Balance the chemical equations in Exercises 6–11 using the vector e...
- 1.6.7E: Balance the chemical equations in Exercises 6–11 using the vector e...
- 1.6.8E: Limestone, CaCO3; neutralizes the acid, H3O, in acid rain by the fo...
- 1.6.9E: Balance the chemical equations in Exercises 6–11 using the vector e...
- 1.6.10E: Balance the chemical equations in Exercises 6–11 using the vector e...
- 1.6.11E: Balance the chemical equations in Exercises 6–11 using the vector e...
- 1.6.12E: Find the general flow pattern of the network shown in the figure. A...
- 1.6.13E: a. Find the general flow pattern of the network shown in the figure...
- 1.6.14E: a. Find the general traffic pattern of the freeway network shown in...
- 1.6.15E: Intersections in England are often constructed as one-way “roundabo...
Solutions for Chapter 1.6: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications | 4th Edition
peA) = det(A - AI) has peA) = zero 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 n-l - An).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
= Xl (column 1) + ... + xn(column n) = combination of columns.
Outer product uv T
= column times row = rank one matrix.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.