- 3.1: Create the matrix A and the demand vector for the open Leontief mod...
- 3.2: Calculate the production vector for the open Leontief model from Ta...
- 3.3: Now suppose that the demand vector is changed to Notice that the on...
- 3.4: Compute You should find that the difference in the production vecto...
- 3.5: How much would the service production level need to increase if dem...
- 3.6: How much would the manufacturing production level need to increase ...
- 3.7: Find the consumption matrix A and demand vector for this table.
- 3.8: Find the production vector X for the consumption matrix A and deman...
- 3.9: If the demand for construction increases by 1 million, how much wil...
- 3.10: Which three sectors are most affected by an increase of $1 million ...
- 3.11: If the entire jth column of the matrix is added, the result is the ...
- 3.12: What would it mean for the (i, j) entry in to be zero?
Solutions for Chapter 3: Matrices
Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach | 11th Edition
Tv = Av + Vo = linear transformation plus shift.
Upper triangular systems are solved in reverse order Xn to Xl.
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.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Invert A by row operations on [A I] to reach [I A-I].
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).
Outer product uv T
= column times row = rank one matrix.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).