 3.4.1: For 18, evaluate the given determinant.3 .
 3.4.2: For 18, evaluate the given determinant.
 3.4.3: For 18, evaluate the given determinant.35 712 46 3 2
 3.4.4: For 18, evaluate the given determinant.
 3.4.5: For 18, evaluate the given determinant.2.3 1.5 7.94.2 3.3 5.16.8 3....
 3.4.6: For 18, evaluate the given determinant.abcbcacab
 3.4.7: For 18, evaluate the given determinant.3 5 1 22 1 523 2 571 1 21
 3.4.8: For 18, evaluate the given determinant.7 1232 2463 15418 9 27 54
 3.4.9: For 914, find det(A). If A is invertible, use the adjoint method to...
 3.4.10: For 914, find det(A). If A is invertible, use the adjoint method to...
 3.4.11: For 914, find det(A). If A is invertible, use the adjoint method to...
 3.4.12: For 914, find det(A). If A is invertible, use the adjoint method to...
 3.4.13: For 914, find det(A). If A is invertible, use the adjoint method to...
 3.4.14: For 914, find det(A). If A is invertible, use the adjoint method to...
 3.4.15: For 1520, use Cramers rule to determine the unique solution for x t...
 3.4.16: For 1520, use Cramers rule to determine the unique solution for x t...
 3.4.17: For 1520, use Cramers rule to determine the unique solution for x t...
 3.4.18: For 1520, use Cramers rule to determine the unique solution for x t...
 3.4.19: For 1520, use Cramers rule to determine the unique solution for x t...
 3.4.20: For 1520, use Cramers rule to determine the unique solution for x t...
 3.4.21: If A is an invertible n n matrix, prove property P9: det(A1) = 1 de...
 3.4.22: If A is an arbitrary 33 matrix, use cofactor expansion to show that...
 3.4.23: For 2329, assume that A and B be 33 matrices with det(A) = 3 and de...
 3.4.24: For 2329, assume that A and B be 33 matrices with det(A) = 3 and de...
 3.4.25: For 2329, assume that A and B be 33 matrices with det(A) = 3 and de...
 3.4.26: For 2329, assume that A and B be 33 matrices with det(A) = 3 and de...
 3.4.27: For 2329, assume that A and B be 33 matrices with det(A) = 3 and de...
 3.4.28: For 2329, assume that A and B be 33 matrices with det(A) = 3 and de...
 3.4.29: For 2329, assume that A and B be 33 matrices with det(A) = 3 and de...
Solutions for Chapter 3.4: Summary of Determinants
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 3.4: Summary of Determinants
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 29 problems in chapter 3.4: Summary of Determinants have been answered, more than 20153 students have viewed full stepbystep solutions from this chapter. Chapter 3.4: Summary of Determinants includes 29 full stepbystep solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.