 3.6.1P: Use cofactor expansions to evaluate the determinants in Problem. Ex...
 3.6.2P: Use cofactor expansions to evaluate the determinants in Problem. Ex...
 3.6.3P: Use cofactor expansions to evaluate the determinants in Problem. Ex...
 3.6.4P: Use cofactor expansions to evaluate the determinants in Problem. Ex...
 3.6.5P: Use cofactor expansions to evaluate the determinants in Problem. Ex...
 3.6.6P: Use cofactor expansions to evaluate the determinants in Problem. Ex...
 3.6.7P: In Problem, evaluate each given determinant after first simplifying...
 3.6.8P: In Problem, evaluate each given determinant after first simplifying...
 3.6.9P: In Problem, evaluate each given determinant after first simplifying...
 3.6.10P: In Problem, evaluate each given determinant after first simplifying...
 3.6.11P: In Problem, evaluate each given determinant after first simplifying...
 3.6.12P: In Problem, evaluate each given determinant after first simplifying...
 3.6.13P: Use the method of elimination to evaluate the determinants in Problem
 3.6.14P: Use the method of elimination to evaluate the determinants in Problem
 3.6.15P: Use the method of elimination to evaluate the determinants in Problem
 3.6.16P: Use the method of elimination to evaluate the determinants in Problem
 3.6.17P: Use the method of elimination to evaluate the determinants in Problem
 3.6.18P: Use the method of elimination to evaluate the determinants in Problem
 3.6.19P: Use the method of elimination to evaluate the determinants in Problem
 3.6.20P: Use the method of elimination to evaluate the determinants in Problem
 3.6.21P: Use Cramer’s rule to solve the systems in Problem.3x + 4y = 25x + 7...
 3.6.22P: Use Cramer’s rule to solve the systems in Problem.5x + 8y = 38x + 1...
 3.6.23P: Use Cramer’s rule to solve the systems in Problem.17x + 7y = 612x +...
 3.6.24P: Use Cramer’s rule to solve the systems in Problem.11x + 15y = 108x ...
 3.6.25P: Use Cramer’s rule to solve the systems in Problem.5x + 6y = 123x + ...
 3.6.26P: Use Cramer’s rule to solve the systems in Problem.6x + 7y = 38x + 9...
 3.6.27P: Use Cramer’s rule to solve the systems in Problem.5x1 + 2x2 ? 2x3 =...
 3.6.28P: Use Cramer’s rule to solve the systems in Problem.5x1 + 4x2 ? 2x3 =...
 3.6.29P: Use Cramer’s rule to solve the systems in Problem.
 3.6.30P: Use Cramer’s rule to solve the systems in Problem.x1 + 4x2 + 2x3 = ...
 3.6.31P: Use Cramer’s rule to solve the systems in Problem.2x1 ? 5x3 = ?34x1...
 3.6.32P: Use Cramer’s rule to solve the systems in Problem.3x1 + 4x2 ? 3x3 =...
 3.6.33P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in .
 3.6.34P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in P...
 3.6.35P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in P...
 3.6.36P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in P...
 3.6.37P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in P...
 3.6.38P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in P...
 3.6.39P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in P...
 3.6.40P: Apply Theorem 5 to find the inverse A?1 of each matrix A given in P...
 3.6.41P: Show that (AB) T = BT AT if A and B are arbitrary 2 × 2 matrices.
 3.6.42P: Consider the 2 × 2 matrices and where x and y denote the row vector...
 3.6.43P: In the a special case of one of Property 1 through Property 5. Veri...
 3.6.44P: In the a special case of one of Property 1 through Property 5. Veri...
 3.6.45P: In the a special case of one of Property 1 through Property 5. Veri...
 3.6.46P: In the a special case of one of Property 1 through Property 5. Veri...
 3.6.47P: Suppose that A and B are matrices of the same size. Show that:(a) (...
 3.6.49P: Let A = [ aij ] be a 3 × 3 matrix. Show that det(AT) = det A by exp...
 3.6.50P: Suppose that A2 = A. Prove that A = 0 or A = 1.
 3.6.51P: Suppose that An = 0 (the zero matrix) for some positive integer n. ...
 3.6.52P: The square matrix A is called orthogonal provided that AT = A?1. Sh...
 3.6.53P: The matrices A and B are said to be similar provided that A = P?1BP...
 3.6.54P: Deduce from Theorems 2 and 3 that if A and B are a n × n invertible...
 3.6.55P: Let A and B be n × n matrices. Suppose it is known that either AB =...
 3.6.56P: Let A be an n × n matrix with det A = 1 and with all elements of A ...
 3.6.59P: Show that and
 3.6.61P: , deal with the Vandermonde determinant Show by direct computation ...
 3.6.64P: , deal with the Vandermonde determinant Use the formula to evaluate...
Solutions for Chapter 3.6: Differential Equations and Linear Algebra 3rd Edition
Full solutions for Differential Equations and Linear Algebra  3rd Edition
ISBN: 9780136054252
Solutions for Chapter 3.6
Get Full SolutionsDifferential Equations and Linear Algebra was written by and is associated to the ISBN: 9780136054252. Chapter 3.6 includes 58 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 58 problems in chapter 3.6 have been answered, more than 12750 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.