- 3.4.1: In 14, find the relative wages of each person for the given closed ...
- 3.4.2: In 14, find the relative wages of each person for the given closed ...
- 3.4.3: In 14, find the relative wages of each person for the given closed ...
- 3.4.4: In 14, find the relative wages of each person for the given closed ...
- 3.4.5: For the three industries R, S, and T in the open Leontief model of ...
- 3.4.6: Rework if the forecast demand vector is D4 = C 100 80 60 S
- 3.4.7: Closed Leontief Model A society consists of four individuals: a far...
- 3.4.8: Closed Leontief Model If in the meat production utilization changes...
- 3.4.9: Open Leontief Model Suppose the interrelationships between the prod...
- 3.4.10: Open Leontief Model Suppose the interrelationships between the prod...
- 3.4.11: Closed Leontief Model In a professional cooperative, a physician, a...
- 3.4.12: Open Leontief Model Suppose three corporations each manufacture a d...
- 3.4.13: Medicine The number of persons in the United States with diagnosed ...
- 3.4.14: Baseball Salaries The average annual salary for major league baseba...
- 3.4.15: In 116, use the following matrices to compute each expression. Stat...
- 3.4.16: In 116, use the following matrices to compute each expression. Stat...
- 3.4.17: In 1722, find the inverse, if it exists, of each matrix.3 0-2 1
- 3.4.18: In 1722, find the inverse, if it exists, of each matrix.4 13 1
- 3.4.19: In 1722, find the inverse, if it exists, of each matrix.4 26 3
- 3.4.20: In 1722, find the inverse, if it exists, of each matrix.123245356
- 3.4.21: In 1722, find the inverse, if it exists, of each matrix.C S4 3 -102...
- 3.4.22: In 1722, find the inverse, if it exists, of each matrix.1 2 -3462-1...
- 3.4.23: What must be true about x, y,z, if we require for the matrices
- 3.4.24: Let with and let Find t such tha
- 3.4.25: Cryptography Use the correspondence A B C D E F G H I J K L M N O P...
- 3.4.26: Ordering Coffee Bill is in charge of ordering coffee for three coff...
- 3.4.27: Stock Purchases On February 3, 2010, Microsoft stock sold for $28.5...
- 3.4.28: Mutual Fund Investing Over the previous 5 years, the median return ...
Solutions for Chapter 3.4: Applications in Economics (the Leontief Model), Accounting, and Statistics (the Method of Least Squares)*
Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach | 11th Edition
Solutions for Chapter 3.4: Applications in Economics (the Leontief Model), Accounting, and Statistics (the Method of Least Squares)*Get Full 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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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.
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.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).