 1.5.1: In Exercises 12, determine whether the given matrix is elementary.(...
 1.5.2: In Exercises 12, determine whether the given matrix is elementary.(...
 1.5.3: In Exercises 34, find a row operation and the corresponding element...
 1.5.4: In Exercises 34, find a row operation and the corresponding element...
 1.5.5: In Exercises 56 an elementary matrix E and a matrix A are given. Id...
 1.5.6: In Exercises 56 an elementary matrix E and a matrix A are given. Id...
 1.5.7: In Exercises 78, use the following matrices and find an elementary ...
 1.5.8: In Exercises 78, use the following matrices and find an elementary ...
 1.5.9: In Exercises 910, first use Theorem 1.4.5 and then use the inversio...
 1.5.10: In Exercises 910, first use Theorem 1.4.5 and then use the inversio...
 1.5.11: In Exercises 1112, use the inversion algorithm to find the inverse ...
 1.5.12: In Exercises 1112, use the inversion algorithm to find the inverse ...
 1.5.13: In Exercises 1318, use the inversion algorithm to find the inverse ...
 1.5.14: In Exercises 1318, use the inversion algorithm to find the inverse ...
 1.5.15: In Exercises 1318, use the inversion algorithm to find the inverse ...
 1.5.16: In Exercises 1318, use the inversion algorithm to find the inverse ...
 1.5.17: In Exercises 1318, use the inversion algorithm to find the inverse ...
 1.5.18: In Exercises 1318, use the inversion algorithm to find the inverse ...
 1.5.19: In Exercises 1920, find the inverse of each of the following 4 4 ma...
 1.5.20: In Exercises 1920, find the inverse of each of the following 4 4 ma...
 1.5.21: In Exercises 2122, find all values of c, if any, for which the give...
 1.5.22: In Exercises 2122, find all values of c, if any, for which the give...
 1.5.23: In Exercises 2326, express the matrix and its inverse as products o...
 1.5.24: In Exercises 2326, express the matrix and its inverse as products o...
 1.5.25: In Exercises 2326, express the matrix and its inverse as products o...
 1.5.26: In Exercises 2326, express the matrix and its inverse as products o...
 1.5.27: In Exercises 2728, show that the matrices A and B are row equivalen...
 1.5.28: In Exercises 2728, show that the matrices A and B are row equivalen...
 1.5.29: Show that if A = 100 010 abc is an elementary matrix, then at least...
 1.5.30: Show that A = 0 a 000 b 0 c 0 0 0 d 0 e 0 0 0 f 0 g 000 h 0 is not ...
 1.5.31: Prove that if A and B are m n matrices, then A and B are row equiva...
 1.5.32: Prove that if A is an invertible matrix and B is row equivalent to ...
 1.5.33: Prove that if B is obtained from A by performing a sequence of elem...
Solutions for Chapter 1.5: Elementary Matrices and a Method for Finding A1
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 1.5: Elementary Matrices and a Method for Finding A1
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. Since 33 problems in chapter 1.5: Elementary Matrices and a Method for Finding A1 have been answered, more than 15006 students have viewed full stepbystep solutions from this chapter. Chapter 1.5: Elementary Matrices and a Method for Finding A1 includes 33 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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 •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.