suppose an nxn matrix has positive diagonal entries and negative everywhere else. suppose sum of the entries in each row are positve. prove det(A) is not 0.
suppose an nxn matrix has positive diagonal entries and negative everywhere else. suppose sum of the entries in each row are positve. prove det(A) is not 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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suppose an nxn matrix has positive diagonal entries and negative everywhere else. suppose sum of the entries in each row are positve. prove det(A) is not 0.
Expert Solution

Step 1
Introduction:
To prove that the determinant of matrix A is non-zero, we will show that A is invertible. If A is invertible, then its determinant is non-zero because det(A) is the product of its eigenvalues, and since all eigenvalues of A are non-zero, det(A) must also be non-zero.
Given:
nxn matrix has positive diagonal entries and negative everywhere else.
We need to prove that det(A) is not 0.
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