 3.3.1: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.1: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.2: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.2: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.3: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.4: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.4: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.5: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.5: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.6: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.6: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.7: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.7: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.8: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.8: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.9: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.9: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.10: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.10: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.11: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.11: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.12: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.12: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.13: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.13: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.14: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.14: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.15: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.15: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.16: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.16: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.17: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.17: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.18: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.3.18: The Determinant of a Matrix In Exercises 118, find the determinant ...
 3.19: Properties of Determinants In Exercises 1922, determine which prope...
 3.3.19: Properties of Determinants In Exercises 1922, determine which prope...
 3.20: Properties of Determinants In Exercises 1922, determine which prope...
 3.3.20: Properties of Determinants In Exercises 1922, determine which prope...
 3.21: Properties of Determinants In Exercises 1922, determine which prope...
 3.3.21: Properties of Determinants In Exercises 1922, determine which prope...
 3.22: Properties of Determinants In Exercises 1922, determine which prope...
 3.3.22: Properties of Determinants In Exercises 1922, determine which prope...
 3.23: The Determinant of a Matrix Product In Exercises 23 and 24, find (a...
 3.3.23: The Determinant of a Matrix Product In Exercises 23 and 24, find (a...
 3.24: The Determinant of a Matrix Product In Exercises 23 and 24, find (a...
 3.3.24: The Determinant of a Matrix Product In Exercises 23 and 24, find (a...
 3.25: Finding Determinants In Exercises 25 and 26, find (a) AT , (b) A3 ,...
 3.3.25: Finding Determinants In Exercises 25 and 26, find (a) AT , (b) A3 ,...
 3.26: Finding Determinants In Exercises 25 and 26, find (a) AT , (b) A3 ,...
 3.3.26: Finding Determinants In Exercises 25 and 26, find (a) AT , (b) A3 ,...
 3.27: Finding Determinants In Exercises 27 and 28, find (a) A and (b) A1 ...
 3.3.27: Finding Determinants In Exercises 27 and 28, find (a) A and (b) A1 ...
 3.28: Finding Determinants In Exercises 27 and 28, find (a) A and (b) A1 ...
 3.3.28: Finding Determinants In Exercises 27 and 28, find (a) A and (b) A1 ...
 3.29: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.3.29: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.30: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.3.30: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.31: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.3.31: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.32: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.3.32: The Determinant of the Inverse of a Matrix In Exercises 2932, find ...
 3.33: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.3.33: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.34: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.3.34: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.35: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.3.35: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.36: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.3.36: Solving a System of Linear Equations In Exercises 3336, solve the s...
 3.37: System of Linear Equations In Exercises 3742, use the determinant o...
 3.3.37: System of Linear Equations In Exercises 3742, use the determinant o...
 3.38: System of Linear Equations In Exercises 3742, use the determinant o...
 3.3.38: System of Linear Equations In Exercises 3742, use the determinant o...
 3.39: System of Linear Equations In Exercises 3742, use the determinant o...
 3.3.39: System of Linear Equations In Exercises 3742, use the determinant o...
 3.40: System of Linear Equations In Exercises 3742, use the determinant o...
 3.3.40: System of Linear Equations In Exercises 3742, use the determinant o...
 3.41: System of Linear Equations In Exercises 3742, use the determinant o...
 3.3.41: System of Linear Equations In Exercises 3742, use the determinant o...
 3.42: System of Linear Equations In Exercises 3742, use the determinant o...
 3.3.42: System of Linear Equations In Exercises 3742, use the determinant o...
 3.43: Let A and B be square matrices of order 4 such thatA = 4 and B = 2....
 3.3.43: Let A and B be square matrices of order 4 such thatA = 4 and B = 2....
 3.44: Let A and B be square matrices of order 3 such thatA = 2 and B = 5....
 3.3.44: Let A and B be square matrices of order 3 such thatA = 2 and B = 5....
 3.45: Proof Prove the property below. a11a21a31 + c31a12a22a32 + c32a13a2...
 3.3.45: Proof Prove the property below. a11a21a31 + c31a12a22a32 + c32a13a2...
 3.46: Illustrate the property in Exercise 45 with A, c31, c32,and c33 bel...
 3.3.46: Illustrate the property in Exercise 45 with A, c31, c32,and c33 bel...
 3.47: Find the determinant of the n n matrix.[1 n1111 n1111. . .. . .. . ...
 3.3.47: Find the determinant of the n n matrix.[1 n1111 n1111. . .. . .. . ...
 3.48: Show thata1111a1111a1111a = (a + 3)(a 1)3.
 3.3.48: Show thata1111a1111a1111a = (a + 3)(a 1)3.
 3.49: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.3.49: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.50: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.3.50: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.51: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.3.51: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.52: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.3.52: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.53: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.3.53: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.54: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.3.54: Calculus In Exercises 4954, find the Jacobians of the functions. If...
 3.55: Writing Compare the various methods for calculating the determinant...
 3.3.55: Writing Compare the various methods for calculating the determinant...
 3.56: Writing Use the table on page 122 to compare the numbers of operati...
 3.3.56: Writing Use the table on page 122 to compare the numbers of operati...
 3.57: Writing Solve the equation for x, if possible. cos xsin xsin x cos ...
 3.3.57: Writing Solve the equation for x, if possible. cos xsin xsin x cos ...
 3.58: Proof Prove that if A = B 0, and A and B are ofthe same size, then ...
 3.3.58: Proof Prove that if A = B 0, and A and B are ofthe same size, then ...
 3.59: Finding the Adjoint of a Matrix In Exercises 59 and 60, find the ad...
 3.3.59: Finding the Adjoint of a Matrix In Exercises 59 and 60, find the ad...
 3.60: Finding the Adjoint of a Matrix In Exercises 59 and 60, find the ad...
 3.3.60: Finding the Adjoint of a Matrix In Exercises 59 and 60, find the ad...
 3.61: System of Linear Equations In Exercises 6164, use the determinant o...
 3.3.61: System of Linear Equations In Exercises 6164, use the determinant o...
 3.62: System of Linear Equations In Exercises 6164, use the determinant o...
 3.3.62: System of Linear Equations In Exercises 6164, use the determinant o...
 3.63: System of Linear Equations In Exercises 6164, use the determinant o...
 3.3.63: System of Linear Equations In Exercises 6164, use the determinant o...
 3.64: System of Linear Equations In Exercises 6164, use the determinant o...
 3.3.64: System of Linear Equations In Exercises 6164, use the determinant o...
 3.65: Using Cramers Rule In Exercises 65 and 66, use a software program o...
 3.3.65: Using Cramers Rule In Exercises 65 and 66, use a software program o...
 3.66: Using Cramers Rule In Exercises 65 and 66, use a software program o...
 3.3.66: Using Cramers Rule In Exercises 65 and 66, use a software program o...
 3.67: Finding the Area of a Triangle In Exercises 67 and 68, use a determ...
 3.3.67: Finding the Area of a Triangle In Exercises 67 and 68, use a determ...
 3.68: Finding the Area of a Triangle In Exercises 67 and 68, use a determ...
 3.3.68: Finding the Area of a Triangle In Exercises 67 and 68, use a determ...
 3.69: Finding an Equation of a Line In Exercises 69 and 70, use a determi...
 3.3.69: Finding an Equation of a Line In Exercises 69 and 70, use a determi...
 3.70: Finding an Equation of a Line In Exercises 69 and 70, use a determi...
 3.3.70: Finding an Equation of a Line In Exercises 69 and 70, use a determi...
 3.71: Finding an Equation of a Plane In Exercises 71 and 72, use a determ...
 3.3.71: Finding an Equation of a Plane In Exercises 71 and 72, use a determ...
 3.72: Finding an Equation of a Plane In Exercises 71 and 72, use a determ...
 3.3.72: Finding an Equation of a Plane In Exercises 71 and 72, use a determ...
 3.73: Using Cramers Rule Determine whether CramersRule is used correctly ...
 3.3.73: Using Cramers Rule Determine whether CramersRule is used correctly ...
 3.74: Health Care Expenditures The table shows annual personal health car...
 3.3.74: Health Care Expenditures The table shows annual personal health car...
 3.75: True or False? In Exercises 7578, determine whether each statement ...
 3.3.75: True or False? In Exercises 7578, determine whether each statement ...
 3.76: True or False? In Exercises 7578, determine whether each statement ...
 3.3.76: True or False? In Exercises 7578, determine whether each statement ...
 3.77: True or False? In Exercises 7578, determine whether each statement ...
 3.3.77: True or False? In Exercises 7578, determine whether each statement ...
 3.78: True or False? In Exercises 7578, determine whether each statement ...
 3.3.78: True or False? In Exercises 7578, determine whether each statement ...
Solutions for Chapter 3: Determinants
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 3: Determinants
Get Full SolutionsElementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. Since 156 problems in chapter 3: Determinants have been answered, more than 45025 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3: Determinants includes 156 full stepbystep solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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).

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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 v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.