 8.3.1: In Exercises 16, fill in the blank. 1. A system of equations that i...
 8.3.2: In Exercises 16, fill in the blank. 2. A solution of a system of th...
 8.3.3: In Exercises 16, fill in the blank. 3. The process used to write a ...
 8.3.4: In Exercises 16, fill in the blank. 4. A system of equations is cal...
 8.3.5: In Exercises 16, fill in the blank. 5. Solutions of equations in th...
 8.3.6: In Exercises 16, fill in the blank. 1. A system of equations that i...
 8.3.7: Is a consistent system with exactly one solution independent or dep...
 8.3.8: Is a consistent system with infinitely many solutions independent o...
 8.3.9: In Exercises 912, determine whether each ordered triple is a soluti...
 8.3.10: In Exercises 912, determine whether each ordered triple is a soluti...
 8.3.11: In Exercises 912, determine whether each ordered triple is a soluti...
 8.3.12: In Exercises 912, determine whether each ordered triple is a soluti...
 8.3.13: In Exercises 1318, use backsubstitution to solve the system of line...
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 8.3.17: In Exercises 1318, use backsubstitution to solve the system of line...
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 8.3.19: In Exercises 19 and 20, perform the row operation and write the equ...
 8.3.20: In Exercises 19 and 20, perform the row operation and write the equ...
 8.3.21: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.22: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.23: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.24: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.25: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.26: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.27: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.28: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.29: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.30: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.31: In Exercises 21 42, solve the system of linear equations and check ...
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 8.3.34: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.35: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.36: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.37: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.38: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.39: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.40: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.41: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.42: In Exercises 21 42, solve the system of linear equations and check ...
 8.3.43: In Exercises 4348, find a system of linear equations that has the g...
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 8.3.49: In Exercises 4952, sketch the plane represented by the linear equat...
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 8.3.53: In Exercises 5360, write the form of the partial fraction decomposi...
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 8.3.61: In Exercises 6174, write the partial fraction decomposition for the...
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 8.3.67: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.68: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.69: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.70: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.71: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.72: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.73: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.74: In Exercises 6174, write the partial fraction decomposition for the...
 8.3.75: In Exercises 75 and 76, write the partial fraction decomposition fo...
 8.3.76: In Exercises 75 and 76, write the partial fraction decomposition fo...
 8.3.77: In Exercises 7780, an object moving vertically is at the given heig...
 8.3.78: In Exercises 7780, an object moving vertically is at the given heig...
 8.3.79: In Exercises 7780, an object moving vertically is at the given heig...
 8.3.80: In Exercises 7780, an object moving vertically is at the given heig...
 8.3.81: In Exercises 81 84, find the equation of the parabola y = ax2 + bx ...
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 8.3.83: In Exercises 81 84, find the equation of the parabola y = ax2 + bx ...
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 8.3.85: In Exercises 8588, find the equation of the circle x2 + y2 + Dx + E...
 8.3.86: In Exercises 8588, find the equation of the circle x2 + y2 + Dx + E...
 8.3.87: In Exercises 8588, find the equation of the circle x2 + y2 + Dx + E...
 8.3.88: In Exercises 8588, find the equation of the circle x2 + y2 + Dx + E...
 8.3.89: A college student borrowed $30,000 to pay for tuition, room, and bo...
 8.3.90: A small corporation borrowed $775,000 to expand its software line. ...
 8.3.91: In Exercises 91 and 92, consider an investor with a portfolio total...
 8.3.92: In Exercises 91 and 92, consider an investor with a portfolio total...
 8.3.93: In the 2013 Womens NCAA Championship basketball game, the Universit...
 8.3.94: The Augusta National Golf Club in Augusta, Georgia, is an 18hole c...
 8.3.95: When Kirchhoffs Laws are applied to the electrical network in the f...
 8.3.96: A system of pulleys is loaded with 128pound and 32pound weights (...
 8.3.97: To find the least squares regression parabola y = ax2 + bx + c for ...
 8.3.98: To find the least squares regression parabola y = ax2 + bx + c for ...
 8.3.99: To find the least squares regression parabola y = ax2 + bx + c for ...
 8.3.100: To find the least squares regression parabola y = ax2 + bx + c for ...
 8.3.101: During the testing of a new automobile braking system, the speeds x...
 8.3.102: A wildlife management team studied the reproduction rates of deer i...
 8.3.103: The predicted cost C (in thousands of dollars) for a company to rem...
 8.3.104: The magnitude of the range R of exhaust temperatures (in degrees Fa...
 8.3.105: In Exercises 105 and 106, determine whether the statement is true o...
 8.3.106: In Exercises 105 and 106, determine whether the statement is true o...
 8.3.107: You are tutoring a student in algebra. In trying to find a partial ...
 8.3.108: Find values of a, b, and c (if possible) such that the system of li...
 8.3.109: Are the two systems of equations equivalent? Give reasons for your ...
 8.3.110: The number of sides x and the combined number of sides and diagonal...
 8.3.111: Find a system of equations in three variables that has exactly two ...
 8.3.112: When using Gaussian elimination to solve a system of linear equatio...
 8.3.113: In Exercises 113 and 114, find values of x, y, and that satisfy the...
 8.3.114: In Exercises 113 and 114, find values of x, y, and that satisfy the...
 8.3.115: In Exercises 115118, (a) determine the real zeros of f and (b) sket...
 8.3.116: In Exercises 115118, (a) determine the real zeros of f and (b) sket...
 8.3.117: In Exercises 115118, (a) determine the real zeros of f and (b) sket...
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 8.3.119: In Exercises 119 and 120, solve the equation. 119. 43 tan 3 = 1 120...
 8.3.120: In Exercises 119 and 120, solve the equation. 119. 43 tan 3 = 1 120...
 8.3.121: To work an extended application analyzing the earnings per share fo...
Solutions for Chapter 8.3: Linear Systems and Matrices
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 8.3: Linear Systems and Matrices
Get Full SolutionsAlgebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. Chapter 8.3: Linear Systems and Matrices includes 121 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Since 121 problems in chapter 8.3: Linear Systems and Matrices have been answered, more than 59010 students have viewed full stepbystep solutions from this chapter.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

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

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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

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.