Help with algebra Invertible matrixes

[boa_co]

hi

Homework Statement

I have to prove or disprove the following statements:

1. Let B be a square matrix nxn. Then there must be an Invertible A such that AB is an upper triangular matrix.
2. Let A be a square Invertible matrix nxn such that the sum of all the entries in each row is equal to one. Then the sum of all the entries in each row of inverse A is also equal to one.

The Attempt at a Solution

Sorry. don't quite know how to approach it. Any hint will be appreciated.

 

[lippyka]

1. This statement is true. A can be chosen to be any upper triangular matrix of the same size as B, since this will be invertible and multiplying it by B will result in an upper triangular matrix. 2. This statement is also true. Since A is invertible, then its inverse A-1 exists, and A-1A = I where I is the nxn identity matrix. Since the sum of all entries in each row of I is 1, then the sum of all entries in each row of A-1 is also equal to one.