 2.2.1: Evaluate each of the following determinants by inspection: (a) ____...
 2.2.2: Let A = 0 1 2 3 1 1 1 1 2 2 3 3 1 2 2 3 (a) Use the elimination met...
 2.2.3: For each of the following, compute the determinant and state whethe...
 2.2.4: Find all possible choices of c that would make the following matrix...
 2.2.5: Let A be an n n matrix and a scalar. Show that det(A) = n det(A)
 2.2.6: Let A be a nonsingular matrix. Show that det(A1) = 1 det(A)
 2.2.7: Let A and B be 33 matrices with det(A) = 4 and det(B) = 5. Find the...
 2.2.8: Show that if E is an elementary matrix, then ET is an elementary ma...
 2.2.9: Let E1, E2, and E3 be 33 elementary matrices of types I, II, and II...
 2.2.10: Let A and B be row equivalent matrices, and suppose that B can be o...
 2.2.11: Let A be an n n matrix. Is it possible for A2+I = O in the case whe...
 2.2.12: Consider the 3 3 Vandermonde matrix V = 1 x1 x2 1 1 x2 x2 2 1 x3 x2...
 2.2.13: Suppose that a 33 matrix A factors into a product 1 0 0 l21 1 0 l31...
 2.2.14: Let A and B be nn matrices. Prove that the product AB is nonsingula...
 2.2.15: Let A and B be n n matrices. Prove that if AB = I , then BA = I . W...
 2.2.16: A matrix A is said to be skew symmetric if AT = A. For example, A =...
 2.2.17: Let A be a nonsingular nn matrix with a nonzero cofactor Ann, and s...
 2.2.18: Let A be a k k matrix and let B be an (n k) (n k) matrix. Let E = I...
 2.2.19: Let A and B be k k matrices and let M = O B A O Show that det(M) = ...
 2.2.20: Show that evaluating the determinant of an n n matrix by cofactors ...
 2.2.21: Show that the elimination method of computing the value of the dete...
Solutions for Chapter 2.2: Properties of Determinants
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 2.2: Properties of Determinants
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 21 problems in chapter 2.2: Properties of Determinants have been answered, more than 5058 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Chapter 2.2: Properties of Determinants includes 21 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8.

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.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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.

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).

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

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

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.

Solvable system Ax = b.
The right side b is in the column space of A.

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

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