 5.1.1: Questions 112 are about the rules for determinants.
 5.1.2: Questions 112 are about the rules for determinants.
 5.1.3: Questions 112 are about the rules for determinants.
 5.1.4: Questions 112 are about the rules for determinants.
 5.1.5: Questions 112 are about the rules for determinants.
 5.1.6: Questions 112 are about the rules for determinants.
 5.1.7: Questions 112 are about the rules for determinants.
 5.1.8: Questions 112 are about the rules for determinants.
 5.1.9: Questions 112 are about the rules for determinants.
 5.1.10: Questions 112 are about the rules for determinants.
 5.1.11: Questions 112 are about the rules for determinants.
 5.1.12: Questions 112 are about the rules for determinants.
 5.1.13: Questions 1327 use the rules to compute specific determinants.
 5.1.14: Questions 1327 use the rules to compute specific determinants.
 5.1.15: Questions 1327 use the rules to compute specific determinants.
 5.1.16: Questions 1327 use the rules to compute specific determinants.
 5.1.17: Questions 1327 use the rules to compute specific determinants.
 5.1.18: Questions 1327 use the rules to compute specific determinants.
 5.1.19: Questions 1327 use the rules to compute specific determinants.
 5.1.20: Questions 1327 use the rules to compute specific determinants.
 5.1.21: Questions 1327 use the rules to compute specific determinants.
 5.1.22: Questions 1327 use the rules to compute specific determinants.
 5.1.23: Questions 1327 use the rules to compute specific determinants.
 5.1.24: Questions 1327 use the rules to compute specific determinants.
 5.1.25: Questions 1327 use the rules to compute specific determinants.
 5.1.26: Questions 1327 use the rules to compute specific determinants.
 5.1.27: Questions 1327 use the rules to compute specific determinants.
 5.1.28: True or false (give a reason if true or a 2 by 2 example if false):...
 5.1.29: What is wrong with this proof that projection matrices have det P =...
 5.1.30: (Calculus question) Show that the partial derivatives ofln(detA) gi...
 5.1.31: (MATLAB) The Hilbert matrix hilb(n) has i, j entry equal to 1/0 + j...
 5.1.32: (MATLAB) What is a typical determinant (experimentally) of rand(n) ...
 5.1.33: (MATLAB) Find the largest determinant of a 6 by 6 matrix of 1 's an...
 5.1.34: If you know that det A = 6, what is the determinant of B? row 1 row...
Solutions for Chapter 5.1: The Properties of Determinants
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 5.1: The Properties of Determinants
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 34 problems in chapter 5.1: The Properties of Determinants have been answered, more than 11659 students have viewed full stepbystep solutions from this chapter. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. Chapter 5.1: The Properties of Determinants includes 34 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4.

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).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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 v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).