 3.3.1: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3.3.1: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.2: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3.3.2: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3.3.3: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3.3.4: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.4: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3.3.5: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.5: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3.3.6: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.6: The Determinant of a Matrix Product In Exercises 16, find (a) A, (b...
 3.3.3.3.7: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.7: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.8: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.8: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.9: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.9: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.10: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.10: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.11: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.11: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.12: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.12: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.13: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.13: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.14: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.14: The Determinant of a Scalar Multiple of a Matrix In Exercises 714, ...
 3.3.3.3.15: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.15: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.3.3.16: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.16: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.3.3.17: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.17: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.3.3.18: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.18: The Determinant of a Matrix Sum In Exercises 1518, find (a) A, (b) ...
 3.3.3.3.19: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.19: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.3.3.20: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.20: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.3.3.21: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.21: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.3.3.22: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.22: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.3.3.23: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.23: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.3.3.24: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.24: Classifying Matrices as Singular or Nonsingular In Exercises 1924, ...
 3.3.3.3.25: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.25: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.3.3.26: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.26: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.3.3.27: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.27: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.3.3.28: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.28: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.3.3.29: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.29: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.3.3.30: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.30: The Determinant of the Inverse of a Matrix In Exercises 2530, find ...
 3.3.3.3.31: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.31: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.3.3.32: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.32: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.3.3.33: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.33: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.3.3.34: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.34: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.3.3.35: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.35: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.3.3.36: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.36: System of Linear Equations In Exercises 3136, use the determinant o...
 3.3.3.3.37: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.37: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.3.3.38: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.38: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.3.3.39: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.39: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.3.3.40: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.40: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.3.3.41: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.41: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.3.3.42: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.42: Singular Matrices In Exercises 3742, find the value(s) of k such th...
 3.3.3.3.43: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.43: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.44: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.44: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.45: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.45: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.46: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.46: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.47: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.47: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.48: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.48: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.49: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.49: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.50: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.50: Finding Determinants In Exercises 4350, find (a) AT , (b) A2 , (c) ...
 3.3.3.3.51: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.51: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.3.3.52: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.52: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.3.3.53: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.53: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.3.3.54: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.54: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.3.3.55: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.55: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.3.3.56: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.56: Finding Determinants In Exercises 5156, use a software program or a...
 3.3.3.3.57: Let A and B be square matrices of order 4 such that A = 5 and B = 3...
 3.3.57: Let A and B be square matrices of order 4 such that A = 5 and B = 3...
 3.3.3.3.58: CAPSTONE Let A and B be square matricesof order 3 such that A = 4 a...
 3.3.58: CAPSTONE Let A and B be square matricesof order 3 such that A = 4 a...
 3.3.3.3.59: Proof Let A and B be n n matrices such thatAB = I. Prove that A 0 a...
 3.3.59: Proof Let A and B be n n matrices such thatAB = I. Prove that A 0 a...
 3.3.3.3.60: Proof Let A and B be n n matrices such that ABis singular. Prove th...
 3.3.60: Proof Let A and B be n n matrices such that ABis singular. Prove th...
 3.3.3.3.61: Find two 2 2 matrices such that A + B = A + B.
 3.3.61: Find two 2 2 matrices such that A + B = A + B.
 3.3.3.3.62: Verify the equation.a + baaaa + baaaa + b = b2(3a + b)
 3.3.62: Verify the equation.a + baaaa + baaaa + b = b2(3a + b)
 3.3.3.3.63: Let A be an n n matrix in which the entries of eachrow sum to zero....
 3.3.63: Let A be an n n matrix in which the entries of eachrow sum to zero....
 3.3.3.3.64: Illustrate the result of Exercise 63 with the matrixA = [ 230112122].
 3.3.64: Illustrate the result of Exercise 63 with the matrixA = [ 230112122].
 3.3.3.3.65: Guided Proof Prove that the determinant of aninvertible matrix A is...
 3.3.65: Guided Proof Prove that the determinant of aninvertible matrix A is...
 3.3.3.3.66: Guided Proof Prove Theorem 3.9: If A is a squarematrix, then det(A)...
 3.3.66: Guided Proof Prove Theorem 3.9: If A is a squarematrix, then det(A)...
 3.3.3.3.67: Writing Let A and P be n n matrices, where P isinvertible. Does P1A...
 3.3.67: Writing Let A and P be n n matrices, where P isinvertible. Does P1A...
 3.3.3.3.68: Writing Let A be an n n nonzero matrixsatisfying A10 = O. Explain w...
 3.3.68: Writing Let A be an n n nonzero matrixsatisfying A10 = O. Explain w...
 3.3.3.3.69: Proof A square matrix is skewsymmetric whenAT = A. Prove that if A...
 3.3.69: Proof A square matrix is skewsymmetric whenAT = A. Prove that if A...
 3.3.3.3.70: Proof Let A be a skewsymmetric matrix of odd order.Use the result ...
 3.3.70: Proof Let A be a skewsymmetric matrix of odd order.Use the result ...
 3.3.3.3.71: True or False? In Exercises 71 and 72, determine whether each state...
 3.3.71: True or False? In Exercises 71 and 72, determine whether each state...
 3.3.3.3.72: True or False? In Exercises 71 and 72, determine whether each state...
 3.3.72: True or False? In Exercises 71 and 72, determine whether each state...
 3.3.3.3.73: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.73: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.3.3.74: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.74: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.3.3.75: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.75: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.3.3.76: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.76: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.3.3.77: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.77: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.3.3.78: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.78: Orthogonal Matrices In Exercises 7378, determine whether the matrix...
 3.3.3.3.79: Proof Prove that the n n identity matrix is orthogonal.
 3.3.79: Proof Prove that the n n identity matrix is orthogonal.
 3.3.3.3.80: Proof Prove that if A is an orthogonal matrix, thenA = 1.
 3.3.80: Proof Prove that if A is an orthogonal matrix, thenA = 1.
 3.3.3.3.81: Orthogonal Matrices In Exercises 81 and 82, use a graphing utility ...
 3.3.81: Orthogonal Matrices In Exercises 81 and 82, use a graphing utility ...
 3.3.3.3.82: Orthogonal Matrices In Exercises 81 and 82, use a graphing utility ...
 3.3.82: Orthogonal Matrices In Exercises 81 and 82, use a graphing utility ...
 3.3.3.3.83: Proof If A is an idempotent matrix (A2 = A), then prove that the de...
 3.3.83: Proof If A is an idempotent matrix (A2 = A), then prove that the de...
 3.3.3.3.84: Proof Let S be an n n singular matrix. Prove that for any n n matri...
 3.3.84: Proof Let S be an n n singular matrix. Prove that for any n n matri...
Solutions for Chapter 3.3: Properties of Determinants
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 3.3: Properties of Determinants
Get Full SolutionsChapter 3.3: Properties of Determinants includes 168 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Since 168 problems in chapter 3.3: Properties of Determinants have been answered, more than 47495 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!

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.