faced with the following coefficient matrix of a system of equations to be solved by iterative methods, 10 -1 -3 -2 33 -7 -6 -3 -27. do you have. any guarantee that you will obtain a solution? cannot tell because the matrix is not symmetric there are no guarantees in life O can't determine because the off-diagonal coefficients are all negative O yes, because the matrix is diagonally dominant O no, because the matrix is not diagonally dominant
faced with the following coefficient matrix of a system of equations to be solved by iterative methods, 10 -1 -3 -2 33 -7 -6 -3 -27. do you have. any guarantee that you will obtain a solution? cannot tell because the matrix is not symmetric there are no guarantees in life O can't determine because the off-diagonal coefficients are all negative O yes, because the matrix is diagonally dominant O no, because the matrix is not diagonally dominant
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter1: Systems Of Linear Equations
Section1.2: Guassian Eliminationa And Guass-jordan Elimination
Problem 67E
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Transcribed Image Text:Question 1
faced with the following coefficient matrix of a system of equations to be solved by iterative methods,
10
-1
-3
-2 33
-7
-6 -3 -27
do you have any guarantee that you will obtain a solution?
cannot tell because the matrix is not symmetric
there are no guarantees in life
O can't determine because the off-diagonal coefficients are all negative
O yes, because the matrix is diagonally dominant
O no, because the matrix is not diagonally dominant
Expert Solution
Step 1
A square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if
where aij denotes the entry in the ith row and jth column.
If a strict inequality (>) is used, this is called strict diagonal dominance.
The Jacobi and Gauss–Seidel methods for solving a linear system converge if the matrix is strictly (or irreducibly) diagonally dominant.
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