 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 oneway “roundabo...
Solutions for Chapter 1.6: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 1.6
Get Full SolutionsChapter 1.6 includes 15 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 15 problems in chapter 1.6 have been answered, more than 35666 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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).

Complex conjugate
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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Graph G.
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 edgenode 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 .

Multiplication Ax
= 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.

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.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI 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.