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.

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.