 6.1.1: Fill in each blank so that the resulting statement is true.A rectan...
 6.1.2: Fill in each blank so that the resulting statement is true.Consider...
 6.1.3: Fill in each blank so that the resulting statement is true.The augm...
 6.1.4: Fill in each blank so that the resulting statement is true.Using Ga...
 6.1.5: Fill in each blank so that the resulting statement is true.True or ...
 6.1.6: Fill in each blank so that the resulting statement is true.True or ...
 6.1.7: In Exercises 18, write the augmented matrix for each system oflinea...
 6.1.8: In Exercises 18, write the augmented matrix for each system oflinea...
 6.1.9: In Exercises 912, write the system of linear equations representedb...
 6.1.10: In Exercises 912, write the system of linear equations representedb...
 6.1.11: In Exercises 912, write the system of linear equations representedb...
 6.1.12: In Exercises 912, write the system of linear equations representedb...
 6.1.13: In Exercises 1318, perform each matrix row operation and writethe n...
 6.1.14: In Exercises 1318, perform each matrix row operation and writethe n...
 6.1.15: In Exercises 1318, perform each matrix row operation and writethe n...
 6.1.16: In Exercises 1318, perform each matrix row operation and writethe n...
 6.1.17: In Exercises 1318, perform each matrix row operation and writethe n...
 6.1.18: In Exercises 1318, perform each matrix row operation and writethe n...
 6.1.19: In Exercises 1920, a few steps in the process of simplifying thegiv...
 6.1.20: In Exercises 1920, a few steps in the process of simplifying thegiv...
 6.1.21: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.22: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.23: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.24: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.25: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.26: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.27: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.28: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.29: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.30: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.31: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.32: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.33: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.34: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.35: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.36: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.37: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.38: In Exercises 2138, solve each system of equations using matrices.Us...
 6.1.39: Find the quadratic function f(x) = ax2 + bx + c for whichf(2) = 4...
 6.1.40: Find the quadratic function f(x) = ax2 + bx + c for whichf(1) = 5,...
 6.1.41: Find the cubic function f(x) = ax3 + bx2 + cx + d forwhich f(1) = ...
 6.1.42: Find the cubic function f(x) = ax3 + bx2 + cx + d forwhich f(1) = ...
 6.1.43: Solve the system:d2 ln w + ln x + 3 ln y  2 ln z = 64 ln w + 3 ln...
 6.1.44: Solve the system:dln w + ln x + ln y + ln z = 1ln w + 4 ln x + ln...
 6.1.45: A ball is thrown straight upward. A position functions(t) = 12 at2 ...
 6.1.46: A football is kicked straight upward. A position functions(t) = 12 ...
 6.1.47: Write a system of linear equations in three or four variables tosol...
 6.1.48: Write a system of linear equations in three or four variables tosol...
 6.1.49: Write a system of linear equations in three or four variables tosol...
 6.1.50: Write a system of linear equations in three or four variables tosol...
 6.1.51: What is a matrix?
 6.1.52: Describe what is meant by the augmented matrix of a systemof linear...
 6.1.53: In your own words, describe each of the three matrix rowoperations....
 6.1.54: Describe how to use row operations and matrices to solve asystem of...
 6.1.55: What is the difference between Gaussian elimination andGaussJordan...
 6.1.56: Most graphing utilities can perform row operations onmatrices. Cons...
 6.1.57: If your graphing utility has a ref (rowechelon form)command or a r...
 6.1.58: Solve using a graphing utilitys ref or rref command:e2x1  2x2 + 3x...
 6.1.59: In Exercises 5962, determine whether eachstatement makes sense or d...
 6.1.60: In Exercises 5962, determine whether eachstatement makes sense or d...
 6.1.61: In Exercises 5962, determine whether eachstatement makes sense or d...
 6.1.62: In Exercises 5962, determine whether eachstatement makes sense or d...
 6.1.63: In Exercises 6366, determine whether each statement is true orfalse...
 6.1.64: In Exercises 6366, determine whether each statement is true orfalse...
 6.1.65: In Exercises 6366, determine whether each statement is true orfalse...
 6.1.66: In Exercises 6366, determine whether each statement is true orfalse...
 6.1.67: The table shows the daily production level and profit for abusiness...
 6.1.68: Solve the system: ex  y = 2y2 = 4x + 4.(Section 5.4, Example 1)
 6.1.69: Graph the solution set of the system:bx + y 7x + 4y 7 8.(Section 5...
 6.1.70: Write as a single logarithm:3 logb x  2 logb 5  13logb y.(Section...
 6.1.71: Exercises 7173 will help you prepare for the material covered inthe...
 6.1.72: Exercises 7173 will help you prepare for the material covered inthe...
 6.1.73: Exercises 7173 will help you prepare for the material covered inthe...
Solutions for Chapter 6.1: Matrix Solutions to Linear Systems
Full solutions for College Algebra  7th Edition
ISBN: 9780134469164
Solutions for Chapter 6.1: Matrix Solutions to Linear Systems
Get Full SolutionsSince 73 problems in chapter 6.1: Matrix Solutions to Linear Systems have been answered, more than 32885 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.1: Matrix Solutions to Linear Systems includes 73 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9780134469164.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

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.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Outer product uv T
= column times row = rank one matrix.

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

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).