Chapter 6.2, Problem 54E

Let A and B be $2\times 2$ matrices with integer entries suchthat $A,A+B,A+2B,A+3B$ , and $A+4B$ are allinvertible matrices whose inverses have integer entries.Show that $A+5B$ is invertible and that its inverse hasinteger entries. This question was in the William LowellPutnam Mathematical Competition in 1994. Hint: Consider the function $f\left(t\right)={\left(\det \left(A+tB\right)\right)}^2-1$ . Show thatthis is a polynomial: what can you say about its degree?Find the values $f\left(0\right),f\left(1\right),f\left(2\right),f\left(3\right),f\left(4\right)$ , usingExercise 53. Now you can determine f(t) by using afamiliar result: If a polynomial $f\left(t\right)$ of degree $\le m$ hasmore than m zeros, then $f\left(t\right)=0$ for all t.