 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 halflife of radium is 1690 years. If 10 gram...
 6.8.10: Radioactive Decay The halflife 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 BoraBora 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
ISBN: 9780321716569
Solutions for Chapter 6.8
Get Full SolutionsAlgebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. Chapter 6.8 includes 25 full stepbystep solutions. Since 25 problems in chapter 6.8 have been answered, more than 57975 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

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).

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 SI AS = A = eigenvalue matrix.

Elimination.
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 IIA1 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, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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.

Multiplier eij.
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 •

Partial pivoting.
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.

Spanning set.
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. AI is also symmetric.