- 6.8.1: Growth of an Insect Population The size P of a certain insect popul...
- 6.8.2: Growth of Bacteria The number N of bacteria present in a culture at...
- 6.8.3: Radioactive Decay Strontium 90 is a radioactive material that decay...
- 6.8.4: Radioactive Decay Iodine 131 is a radioactive material that decays ...
- 6.8.5: Growth of a Colony of Mosquitoes The population of a colony of mosq...
- 6.8.6: Bacterial Growth A culture of bacteria obeys the law of uninhibited...
- 6.8.7: Population Growth The population of a southern city follows the exp...
- 6.8.8: Population Decline The population of a midwestern city follows the ...
- 6.8.9: Radioactive Decay The half-life of radium is 1690 years. If 10 gram...
- 6.8.10: Radioactive Decay The half-life of radioactive potassium is 1.3 bil...
- 6.8.11: Estimating the Age of a Tree A piece of charcoal is found to contai...
- 6.8.12: Estimating the Age of a Fossil A fossilized leaf contains 70% of it...
- 6.8.13: Cooling Time of a Pizza Pan A pizza pan is removed at 5:00 PM from ...
- 6.8.14: Newtons Law of Cooling A thermometer reading 72F is placed in a ref...
- 6.8.15: Newtons Law of Heating A thermometer reading 8C is brought into a r...
- 6.8.16: Warming Time of a Beer Stein A beer stein has a temperature of 28F....
- 6.8.17: Decomposition of Chlorine in a Pool Under certain water conditions,...
- 6.8.18: Decomposition of Dinitrogen Pentoxide At 45C, dinitrogen pentoxide ...
- 6.8.19: Decomposition of Sucrose Reacting with water in an acidic solution ...
- 6.8.20: Decomposition of Salt in Water Salt (NaCl) decomposes in water into...
- 6.8.21: Radioactivity from Chernobyl After the release of radioactive mater...
- 6.8.22: Pig Roasts The hotel Bora-Bora is having a pig roast. At noon, the ...
- 6.8.23: Population of a Bacteria Culture The logistic growth model represen...
- 6.8.24: Population of an Endangered Species Often environmentalists capture...
- 6.8.25: The Challenger Disaster After the Challenger disaster in 1986, a st...
Solutions for Chapter 6.8: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry | 9th 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).
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
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.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.