 3.1.1: For 16, determine the number of inversions and the parity of the gi...
 3.1.2: For 16, determine the number of inversions and the parity of the gi...
 3.1.3: For 16, determine the number of inversions and the parity of the gi...
 3.1.4: For 16, determine the number of inversions and the parity of the gi...
 3.1.5: For 16, determine the number of inversions and the parity of the gi...
 3.1.6: For 16, determine the number of inversions and the parity of the gi...
 3.1.7: Use Definition 3.1.8 to derive the general expression for the deter...
 3.1.8: For 811, determine whether the given expression is a term in the de...
 3.1.9: For 811, determine whether the given expression is a term in the de...
 3.1.10: For 811, determine whether the given expression is a term in the de...
 3.1.11: For 811, determine whether the given expression is a term in the de...
 3.1.12: For 1215, determine the values of the indices p and q such that the...
 3.1.13: For 1215, determine the values of the indices p and q such that the...
 3.1.14: For 1215, determine the values of the indices p and q such that the...
 3.1.15: For 1215, determine the values of the indices p and q such that the...
 3.1.16: For 1642, evaluate the determinant of the given matrix.A =0 25 1.1
 3.1.17: For 1642, evaluate the determinant of the given matrix.
 3.1.18: For 1642, evaluate the determinant of the given matrix.
 3.1.19: For 1642, evaluate the determinant of the given matrix.A =2 31 5
 3.1.20: For 1642, evaluate the determinant of the given matrix.
 3.1.21: For 1642, evaluate the determinant of the given matrix.A = 2 41 0.
 3.1.22: For 1642, evaluate the determinant of the given matrix.
 3.1.23: For 1642, evaluate the determinant of the given matrix.A = e3 3e102...
 3.1.24: For 1642, evaluate the determinant of the given matrix.A = e3 3e102...
 3.1.25: For 1642, evaluate the determinant of the given matrix.A=61 24 710 31
 3.1.26: For 1642, evaluate the determinant of the given matrix.
 3.1.27: For 1642, evaluate the determinant of the given matrix.A=24 16 1121 3
 3.1.28: For 1642, evaluate the determinant of the given matrix.A=0 0304 321 5
 3.1.29: For 1642, evaluate the determinant of the given matrix..A=9 1762 14 02
 3.1.30: For 1642, evaluate the determinant of the given matrix.A=210 31 110 83
 3.1.31: For 1642, evaluate the determinant of the given matrix.A=5 432 9 12...
 3.1.32: For 1642, evaluate the determinant of the given matrix.A=504030201
 3.1.33: For 1642, evaluate the determinant of the given matrix.A=e2e167 1/3...
 3.1.34: For 1642, evaluate the determinant of the given matrix.A=2 3114 131 6
 3.1.35: For 1642, evaluate the determinant of the given matrix.A=00 030 071...
 3.1.36: For 1642, evaluate the determinant of the given matrix.A=4186 0 2 1...
 3.1.37: For 1642, evaluate the determinant of the given matrix.A=20 1 6 1 3...
 3.1.38: For 1642, evaluate the determinant of the given matrix.A=21 460 10 ...
 3.1.39: For 1642, evaluate the determinant of the given matrix.=1200 2 8000...
 3.1.40: For 1642, evaluate the determinant of the given matrix.A=1 23 0 0 2...
 3.1.41: For 1642, evaluate the determinant of the given matrix.A=1200034000...
 3.1.42: For 1642, evaluate the determinant of the given matrix.=0 00 840 00...
 3.1.43: For 4346, evaluate the determinant of the given matrix function.A(t...
 3.1.44: For 4346, evaluate the determinant of the given matrix function.A(t...
 3.1.45: For 4346, evaluate the determinant of the given matrix function.A(t...
 3.1.46: For 4346, evaluate the determinant of the given matrix function.(t)...
 3.1.47: In 4748, we explore a relationship between determinants and solutio...
 3.1.48: In 4748, we explore a relationship between determinants and solutio...
 3.1.49: (a) Write all 24 distinct permutations of the integers 1, 2, 3, 4. ...
 3.1.50: Use to evaluate det(A), where A = 2 0 1 5 0612 1 1 2 3 020 4
 3.1.51: Use to evaluate det(A), where A = 1 4 7 0 30 1 1 21 3 3 02 2 4 .
 3.1.52: Use 49 and 50 to evaluate det(B), where B = 21 5 5 0 02 0 1 5 00 6 ...
 3.1.53: Use 49 and 51 to evaluate det(B), where B = 1 4 700 30 1 1 0 21 3 3...
 3.1.54: (a) If A = a11 a12 a21 a22 and c is a constant, verify that det(cA)...
 3.1.55: The alternating symbol ijk is defined by ijk = 1, if (ijk) is an ev...
 3.1.56: If A is the general n n matrix, determine the sign attached to the ...
 3.1.57: Use some form of technology to evaluate the determinants in 4046.
 3.1.58: Let A be an arbitrary 44 matrix. By experimenting with various elem...
 3.1.59: Verify that y1(x) = e2x cos 3x, y2(x) = e2x sin 3x, and y3(x) = e4x...
 3.1.60: Verify that y1(x) = e2x cos 3x, y2(x) = e2x sin 3x, and y3(x) = e4x...
Solutions for Chapter 3.1: The Definition of the Determinant
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 3.1: The Definition of the Determinant
Get Full SolutionsSince 60 problems in chapter 3.1: The Definition of the Determinant have been answered, more than 19195 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. Differential Equations was written by and is associated to the ISBN: 9780321964670. Chapter 3.1: The Definition of the Determinant includes 60 full stepbystep solutions.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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.

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

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

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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