×
×

# Solutions for Chapter 3.4: Inverse of a Matrix

## Full solutions for Elementary Linear Algebra with Applications | 9th Edition

ISBN: 9780132296540

Solutions for Chapter 3.4: Inverse of a Matrix

Solutions for Chapter 3.4
4 5 0 362 Reviews
30
5
##### ISBN: 9780132296540

Chapter 3.4: Inverse of a Matrix includes 15 full step-by-step solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 15 problems in chapter 3.4: Inverse of a Matrix have been answered, more than 11987 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• 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.

• 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.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• 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.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Simplex method for linear programming.

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

• 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.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B II·

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

×