 8.1.1: In Exercises 14, fill in the blank(s). 1. A set of two or more equa...
 8.1.2: In Exercises 14, fill in the blank(s).2. A _______ of a system of e...
 8.1.3: In Exercises 14, fill in the blank(s). 3. The first step in solving...
 8.1.4: In Exercises 14, fill in the blank(s). 4. A point of intersection o...
 8.1.5: What is the point of intersection of the graphs of the cost and rev...
 8.1.6: The graphs of the equations of a system do not intersect. What can ...
 8.1.7: In Exercises 710, determine whether each ordered pair is a solution...
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 8.1.11: In Exercises 1118, solve the system by the method of substitution. ...
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 8.1.19: In Exercises 1928, solve the system by the method of substitution. ...
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 8.1.27: In Exercises 1928, solve the system by the method of substitution. ...
 8.1.28: In Exercises 1928, solve the system by the method of substitution. ...
 8.1.29: In Exercises 2932, you are given the yearly interest earned from a ...
 8.1.30: In Exercises 2932, you are given the yearly interest earned from a ...
 8.1.31: In Exercises 2932, you are given the yearly interest earned from a ...
 8.1.32: In Exercises 2932, you are given the yearly interest earned from a ...
 8.1.33: In Exercises 3338, solve the system by the method of substitution. ...
 8.1.34: In Exercises 3338, solve the system by the method of substitution. ...
 8.1.35: In Exercises 3338, solve the system by the method of substitution. ...
 8.1.36: In Exercises 3338, solve the system by the method of substitution. ...
 8.1.37: In Exercises 3338, solve the system by the method of substitution. ...
 8.1.38: In Exercises 3338, solve the system by the method of substitution. ...
 8.1.39: In Exercises 3946, solve the system graphically. Verify your soluti...
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 8.1.43: In Exercises 3946, solve the system graphically. Verify your soluti...
 8.1.44: In Exercises 3946, solve the system graphically. Verify your soluti...
 8.1.45: In Exercises 3946, solve the system graphically. Verify your soluti...
 8.1.46: In Exercises 3946, solve the system graphically. Verify your soluti...
 8.1.47: In Exercises 4760, use a graphing utility to approximate all points...
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 8.1.57: In Exercises 4760, use a graphing utility to approximate all points...
 8.1.58: In Exercises 4760, use a graphing utility to approximate all points...
 8.1.59: In Exercises 4760, use a graphing utility to approximate all points...
 8.1.60: In Exercises 4760, use a graphing utility to approximate all points...
 8.1.61: In Exercises 6174, solve the system graphically or algebraically. E...
 8.1.62: In Exercises 6174, solve the system graphically or algebraically. E...
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 8.1.70: In Exercises 6174, solve the system graphically or algebraically. E...
 8.1.71: In Exercises 6174, solve the system graphically or algebraically. E...
 8.1.72: In Exercises 6174, solve the system graphically or algebraically. E...
 8.1.73: In Exercises 6174, solve the system graphically or algebraically. E...
 8.1.74: In Exercises 6174, solve the system graphically or algebraically. E...
 8.1.75: In Exercises 7578, use a graphing utility to graph the cost and rev...
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 8.1.77: In Exercises 7578, use a graphing utility to graph the cost and rev...
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 8.1.79: In Exercises 79 and 80, find the dimensions of the rectangle meetin...
 8.1.80: In Exercises 79 and 80, find the dimensions of the rectangle meetin...
 8.1.81: The daily DVD rentals of a newly released animated film and a newly...
 8.1.82: You want to buy either a wood pellet stove or an electric furnace. ...
 8.1.83: A small software company invests $16,000 to produce a software pack...
 8.1.84: You are offered two jobs selling college textbooks. One company off...
 8.1.85: What are the dimensions of a rectangular tract of land with a perim...
 8.1.86: What are the dimensions of an isosceles right triangle with a twoi...
 8.1.87: You are deciding how to invest a total of $20,000 in two funds payi...
 8.1.88: You are offered two different rules for estimating the number of bo...
 8.1.89: The populations (in thousands) of Colorado C and Minnesota M from 2...
 8.1.90: The table shows the yearly revenues (in millions of dollars) of the...
 8.1.91: In Exercises 91 and 92, determine whether the statement is true or ...
 8.1.92: In Exercises 91 and 92, determine whether the statement is true or ...
 8.1.93: When solving a system of equations by substitution, how do you reco...
 8.1.94: Find equations of lines whose graphs intersect the graph of the par...
 8.1.95: Create systems of two linear equations in two variables that have (...
 8.1.96: Create a system of linear equations in two variables that has the s...
 8.1.97: Consider the system of equations. { y y = = bx xb (a) Use a graphin...
 8.1.98: The cost C of producing x units and the revenue R obtained by selli...
 8.1.99: In Exercises 99102, write an equation of the line passing through t...
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 8.1.103: In Exercises 103108, find the domain of the function and identify a...
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 8.1.107: In Exercises 103108, find the domain of the function and identify a...
 8.1.108: In Exercises 103108, find the domain of the function and identify a...
Solutions for Chapter 8.1: Linear Systems and Matrices
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 8.1: Linear Systems and Matrices
Get Full SolutionsAlgebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.1: Linear Systems and Matrices includes 108 full stepbystep solutions. Since 108 problems in chapter 8.1: Linear Systems and Matrices have been answered, more than 60824 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cyclic shift
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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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).

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.