Solution Found!
We call the vector x Rn integral if every component xi is an integer. Let A be a
Chapter 5, Problem 8(choose chapter or problem)
We call the vector \(\mathbf{x} \in \mathbb{R}^{n}\) integral if every component \(x_i\) is an integer. Let A be a nonsingular \(n \times n\) matrix with integer entries. Prove that the system of equations Ax = b has an integral solution for every integral vector \(\mathbf{b} \in \mathbb{R}^{n}\) if and only if det A = \(\pm 1\). (Note that if A has integer entries, \(\mu_{A}\) maps integral vectors to integral vectors. When does \(\mu_{A}\) map the set of all integral vectors onto the set of all integral vectors?)
Questions & Answers
QUESTION:
We call the vector \(\mathbf{x} \in \mathbb{R}^{n}\) integral if every component \(x_i\) is an integer. Let A be a nonsingular \(n \times n\) matrix with integer entries. Prove that the system of equations Ax = b has an integral solution for every integral vector \(\mathbf{b} \in \mathbb{R}^{n}\) if and only if det A = \(\pm 1\). (Note that if A has integer entries, \(\mu_{A}\) maps integral vectors to integral vectors. When does \(\mu_{A}\) map the set of all integral vectors onto the set of all integral vectors?)
ANSWER:Problem 8
We call the vector
Step by Step Solution
Step 1 of 3
For an 
Now, the solution 
It is already given that
