 5.1: LetA D%2 !3 47 & and B D%3 !4 51 &: Find (a) 2AC3B; (b) 3A!2B; (c) ...
 5.2: Verifythat (a) A.BC/D.AB/C andthat (b) A.BCC/D ABCAC, whereA and B ...
 5.3: Find AB and BA given A D%20 !13 !45 & and B D2 413 !70 3 !2 3 5:
 5.4: Let A and B be the matrices given in and let x D% 2t e!t & and y D2...
 5.5: Let A D2 432 !1 043 !527 3 5 and B D2 40 !32 14 !3 25 !1 3 5: Find ...
 5.6: LetA1 D% 21 !32 &; A2 D% 13 !1 !2&; B D%24 12 &: (a) Show that A1B ...
 5.7: Compute the determinants of the matrices A and B in 6. Are your res...
 5.8: Suppose that A and B are the matrices of 5. Verify that det.AB/ Dde...
 5.9: In 9 and 10, verify the product law for differentiation, .AB/0 D A0...
 5.10: In 9 and 10, verify the product law for differentiation, .AB/0 D A0...
 5.11: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.12: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.13: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.14: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.15: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.16: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.17: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.18: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.19: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.20: In 11 through 20, write the given system in the form x0 D P.t/xCf.t...
 5.21: In 21 through 30, rst verify that the given vectors are solutions o...
 5.22: In 21 through 30, rst verify that the given vectors are solutions o...
 5.23: In 21 through 30, rst verify that the given vectors are solutions o...
 5.24: In 21 through 30, rst verify that the given vectors are solutions o...
 5.25: In 21 through 30, rst verify that the given vectors are solutions o...
 5.26: In 21 through 30, rst verify that the given vectors are solutions o...
 5.27: In 21 through 30, rst verify that the given vectors are solutions o...
 5.28: In 21 through 30, rst verify that the given vectors are solutions o...
 5.29: In 21 through 30, rst verify that the given vectors are solutions o...
 5.30: In 21 through 30, rst verify that the given vectors are solutions o...
 5.31: In 31 through 40, nd a particular solution of the indicated linear ...
 5.32: In 31 through 40, nd a particular solution of the indicated linear ...
 5.33: In 31 through 40, nd a particular solution of the indicated linear ...
 5.34: In 31 through 40, nd a particular solution of the indicated linear ...
 5.35: In 31 through 40, nd a particular solution of the indicated linear ...
 5.36: In 31 through 40, nd a particular solution of the indicated linear ...
 5.37: In 31 through 40, nd a particular solution of the indicated linear ...
 5.38: In 31 through 40, nd a particular solution of the indicated linear ...
 5.39: In 31 through 40, nd a particular solution of the indicated linear ...
 5.40: In 31 through 40, nd a particular solution of the indicated linear ...
 5.41: (a) Show that the vector functions x1.t/ D% t t2& and x2 D%t2 t3& a...
 5.42: Suppose that one of the vector functions x1.t/ D%x11.t/ x21.t/& and...
 5.43: Suppose that the vectors x1.t/ and x2.t/ of are solutions of the eq...
 5.44: Generalize 42 and 43 to prove Theorem 2 for n an arbitrary positive...
 5.45: Let x1.t/, x2.t/, :::;xn.t/ be vector functions whose ith component...
 5.46: In 42 through 50, use a calculator or computer system to calculate ...
 5.47: In 42 through 50, use a calculator or computer system to calculate ...
 5.48: In 42 through 50, use a calculator or computer system to calculate ...
 5.49: In 42 through 50, use a calculator or computer system to calculate ...
 5.50: In 42 through 50, use a calculator or computer system to calculate ...
Solutions for Chapter 5: Linear Systems of Differential Equations
Full solutions for Differential Equations: Computing and Modeling  5th Edition
ISBN: 9780321816252
Solutions for Chapter 5: Linear Systems of Differential Equations
Get Full SolutionsSince 50 problems in chapter 5: Linear Systems of Differential Equations have been answered, more than 1909 students have viewed full stepbystep solutions from this chapter. Differential Equations: Computing and Modeling was written by and is associated to the ISBN: 9780321816252. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations: Computing and Modeling, edition: 5. Chapter 5: Linear Systems of Differential Equations includes 50 full stepbystep solutions.

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).

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

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

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.

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

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.