Answered: suppose an nxn matrix has positive…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter7: Systems Of Equations And Inequalities

Section7.7: Solving Systems With Inverses

Problem 5SE: Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why...

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Question

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.