 2.7.1: Consider the following inputoutput matrix that defines the interde...
 2.7.2: In Exercises 26 consider the economies consisting of either two or...
 2.7.3: In Exercises 26 consider the economies consisting of either two or...
 2.7.4: In Exercises 26 consider the economies consisting of either two or...
 2.7.5: In Exercises 26 consider the economies consisting of either two or...
 2.7.6: In Exercises 26 consider the economies consisting of either two or...
 2.7.7: In Exercises 79 consider the economies consisting of either two or...
 2.7.8: In Exercises 79 consider the economies consisting of either two or...
 2.7.9: In Exercises 79 consider the economies consisting of either two or...
 2.7.10: Let aij be an arbitrary element of an inputoutput matrix. Why woul...
 2.7.11: In an economically feasible situation the sum of the elements of ea...
 2.7.12: Consider a twoindustry economy described by an inputoutput matrixA...
Solutions for Chapter 2.7: The Leontief InputOutput Model in Economics
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 2.7: The Leontief InputOutput Model in Economics
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. Since 12 problems in chapter 2.7: The Leontief InputOutput Model in Economics have been answered, more than 9486 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.7: The Leontief InputOutput Model in Economics includes 12 full stepbystep 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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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 .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

Norm
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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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