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# Solutions for Chapter 3.2: Determinants and Elementary Operations

## Full solutions for Elementary Linear Algebra | 8th Edition

ISBN: 9781305658004

Solutions for Chapter 3.2: Determinants and Elementary Operations

Solutions for Chapter 3.2
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##### ISBN: 9781305658004

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.2: Determinants and Elementary Operations includes 96 full step-by-step solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Since 96 problems in chapter 3.2: Determinants and Elementary Operations have been answered, more than 46925 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• 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.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• 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.

• Jordan form 1 = M- 1 AM.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• 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.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Partial pivoting.

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

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Solvable system Ax = b.

The right side b is in the column space of A.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Volume of box.

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

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