MHB Solving Systems of Equations with Whole Numbers

[Yankel]

Hello all

I have a couple of short questions, both similar, which I do not know how to even start, and I could use some help.

1) Show that if in the system Ax=b, det(A)=-1, and all the members of A are whole numbers (belong to Z), and all the members of b are whole numbers (belong to Z), then a single solution exists, and it's members are also whole numbers.

2) Show that if in the system Ax=b, det(A)=2, and all the members of A are whole, and all the members of b are even, then a single solution exists, and all it's members are whole numbers.

Thank you in advance !

 

[Evgeny.Makarov]

The solution is $x=A^{-1}b$. One way to find $A^{-1}$ is $\frac{1}{\det A}\text{adj}\,A$ where $\text{adj}\,A$ is the adjugate of $A$. By construction, if $A$ consists of whole numbers, then so does $\text{adj}\,A$. In the second case, one can factor out 2 from $b$ and cancel it with $\det A$.