 11.1: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.2: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.3: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.4: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.5: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.6: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.7: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.8: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.9: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.10: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.11: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.12: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.13: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.14: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.15: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.16: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.17: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.18: In Exercises 1 to 18, find the third and seventh terms of the seque...
 11.19: In Exercises 19 to 26, evaluate the expression.
 11.20: In Exercises 19 to 26, evaluate the expression.
 11.21: In Exercises 19 to 26, evaluate the expression.
 11.22: In Exercises 19 to 26, evaluate the expression.
 11.23: In Exercises 19 to 26, evaluate the expression.
 11.24: In Exercises 19 to 26, evaluate the expression.
 11.25: In Exercises 19 to 26, evaluate the expression.
 11.26: In Exercises 19 to 26, evaluate the expression.
 11.27: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.28: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.29: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.30: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.31: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.32: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.33: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.34: In Exercises 27 to 34, find the requested term or partial sum for t...
 11.35: In Exercises 35 and 36, insert the arithmetic means.
 11.36: In Exercises 35 and 36, insert the arithmetic means.
 11.37: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.38: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.39: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.40: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.41: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.42: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.43: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.44: In Exercises 37 to 44, find the requested term or sum for the geome...
 11.45: In Exercises 45 to 48, evaluate the given series.
 11.46: In Exercises 45 to 48, evaluate the given series.
 11.47: In Exercises 45 to 48, evaluate the given series.
 11.48: In Exercises 45 to 48, evaluate the given series.
 11.49: In Exercises 49 and 50, write each number as the ratio of two integ...
 11.50: In Exercises 49 and 50, write each number as the ratio of two integ...
 11.51: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.52: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.53: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.54: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.55: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.56: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.57: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.58: In Exercises 51 to 58, determine whether the sequence is arithmetic...
 11.59: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.60: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.61: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.62: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.63: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.64: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.65: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.66: In Exercises 59 to 66, use mathematical induction to prove each sta...
 11.67: In Exercises 67 to 72, use the Binomial Theorem to expand each bino...
 11.68: In Exercises 67 to 72, use the Binomial Theorem to expand each bino...
 11.69: In Exercises 67 to 72, use the Binomial Theorem to expand each bino...
 11.70: In Exercises 67 to 72, use the Binomial Theorem to expand each bino...
 11.71: In Exercises 67 to 72, use the Binomial Theorem to expand each bino...
 11.72: In Exercises 67 to 72, use the Binomial Theorem to expand each bino...
 11.73: Car Options The buyer of a new car is offered 12 exterior colors an...
 11.74: Dinner Options A restaurant offers a prix fixe dinner that includes...
 11.75: Computer Passwords A computer password consists of eight letters. H...
 11.76: Serial Numbers The serial number on an airplane consists of the let...
 11.77: Committee Membership From a committee of 15 members, a president, a...
 11.78: Arranging Books Three of five different books are to be displayed o...
 11.79: Scheduling The emergency staff at a hospital consists of 4 supervis...
 11.80: Committee Membership From 12 people, a committee of 5 people is for...
 11.81: Playing Cards How many different fourcard hands can be drawn witho...
 11.82: Playing Cards Two cards are drawn, without replacement, from the fo...
 11.83: Number Theory Three numbers are drawn from the digits 1 through 5, ...
 11.84: Tossing Coins A coin is tossed five times. List the elements in the...
 11.85: Dice Two dice are tossed. List the elements in the event that the s...
 11.86: Number Theory Two numbers are drawn, without replacement, from the ...
 11.87: Number Theory Let S = 5Natural numbers less than or equal to 1006 a...
 11.88: Playing Cards A deck of 10 cards contains 5 red and 5 black cards. ...
 11.89: Playing Cards Which of the following has the greater probability: d...
 11.90: Sums of Coins A nickel, a dime, and a quarter are tossed. What is t...
 11.91: Medicine A company claims that its cold remedy is successful in red...
 11.92: Community Government A survey of members in a city council indicate...
 11.93: Employee Badges A room contains 12 employees who are wearing badges...
 11.94: Gordon Model of Stock Valuation Suppose a stock pays a dividend of ...
 11.95: . Multiplier Effect Suppose a city estimates that a new sports faci...
Solutions for Chapter 11: College Algebra and Trigonometry 7th Edition
Full solutions for College Algebra and Trigonometry  7th Edition
ISBN: 9781439048603
Solutions for Chapter 11
Get Full SolutionsSince 95 problems in chapter 11 have been answered, more than 5880 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11 includes 95 full stepbystep solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column space C (A) =
space of all combinations of the columns of A.

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 matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.