6. If A is an nxn matrix whose LU decomposition is PA = LU, prove thatdet(A) = (-1)' u₁U22.²'nn'where r is the number of row interchanges performed during row reduction, and uare the diagonal elements of U.

Given: A is an n×nmatrix, whose LU decomposition is PA=LU. To show: detA=-1ru11u22...unn, where r…

Answered: 6. If A is an nxn matrix whose LU…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 3AEXP

See similar textbooks

Similar questions

Show that A=[0110] has no real eigenvalues.
Solve for the values of x and y using LU Decomposition from the given matrix and prove that PA=LU or A=LU.
A is a 2 X 2 matrix with eigenvectors v1 and v2 corresponding to eigenvalues λ1 = 1/2and λ2 = 2, respectively, and x Find A kx. What happens as k becomes large (i.e., k--> ∞)
Find the general solution of the linear system (1) when A is then x n diagonal matrix A = diag[Aj, A21... ,An]- What condition onthe eigenvalues Al, ... , An will guarantee that limt_,,. x(t) = 0 for allsolutions x(t) of (1)?
Could you do part a of this problem? I'm not sure if I did the Lu and LDU-decompositions correctly?
I don’t understand here how they got the associated matrix like the part with f(1,0,0,0) and so on are they substituting in the coordinates or in alpha or both ?
If the sum of dimensions of all eigenspaces for nxn matrix A is less than n then any vector from Rn can be represented by linearly independent eigenvectors of A. TRUE OR FALSE?
If ldet(A)I > 1, prove that the powers An cannot stay bounded. But if ldet(A)I :S 1, show that some entries of An might still grow large. Eigenvalues will give the right test for stability, determinants tell us only one number.
Suppose the matrix A is used to transform points in the plane iteratively. That is, given a point v, consider the sequence vn = Anv. Letting U = [u1 u2] so that ui is an eigenvector associated to λi and letting v = c1u1 + c2u2 what is a simple expressions for an and bn so that vn = Anv = anu1 + bnu2.
The coefficient matrix is not strictly diagonally dominant, nor can the equations be rearranged to make it so. However, both the Jacobi and the Gauss-Seidel method converge anyway. Demonstrate that this is true of the Gauss-Seidel method, starting with the zero vector as the initial approximation and obtaining a solution that is accurate to within 0.01.
Let A be an n×n matrix with n different positive eigenvalues. Prove that there exists a matrix B such that B2 = A. Is B uniquely determined? Please do this step by step in detail, show why you are allowed to take certain steps, I often have problems finding the right theorem to apply.
Jn is simply an nxn matrix with only ones inside.

SEE MORE QUESTIONS

Recommended textbooks for you

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

6. If A is an nxn matrix whose LU decomposition is PA=LU, prove that det(A)=(-1)' u₁22... Un where r is the number of row interchanges performed during row reduction, and u are the diagonal elements of U.