Show matrix $A+5B$ has an inverse with integer entries given the following conditions

Question

Let $A$ and $B$ be 2×2 matrices with integer entries such that each of $A$, $A + B$, $A + 2B$, $A + 3B$, $A + 4B$ has an inverse with integer entries. Show that the same is true for $A + 5B$.

Answer 1

First note that a matrix $X$ with integer entries is invertible and its inverse has integer entries if and only if $\det(X)=\pm1$.

Let $P(x)=\det(A+xB)$. Then $P(x)$ is a polynomial of degree at most $4$, with integer coefficients, and $P(0),P(1),P(2),P(3),P(4) \in \{ \pm 1 \}$.

Claim: $P(0)=P(1)=P(3)=P(4)$.

Proof: It is known that $b-a|P(b)-P(a)$ for $a,b$ integers.

Then $3|P(4)-P(1), 3|P(3)-P(0)$ and $4|P(4)-P(0)$. Since the RHS of each division is $0$ or $\pm 2$, it follows it is zero. This proves the claim.

Now, $P(x)-P(0)$ is a polynomial of degree at most four which has the roots $0,1,3,4$. Thus

$$P(x)-P(0)=ax(x-1)(x-3)(x-4) \,.$$

hence $P(2)=P(0)-4a$. Since $P(2), P(0) \in \{ \pm 1 \}$, it follows that $a=0$, and hence $P(x)$ is the constant polynomial $1$ or $-1$.

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Extra One can actually deduce further from here that $\det(A)=\pm 1$ and $A^{-1}B$ is nilpotenet.

Indeed $A$ is invertible and since $\det(A+xB)=\det(A)\det(I+xA^{-1}B)$ you get from here that $\det(I+xA^{-1}B)=1$. It is trivial to deduce next that the characteristic polynomial of $A^{-1}B$ is $x^2$.

Answer 2

Hint $\rm\:det(A\!+\!xB)\:$ is a quadratic polynomial with value $\pm1$ at $5$ points so it has value $1$ or $-1$ at $3$ points, so the polynomial must be constant, either $1$ or $-1,\:$ so $\rm\:A\!+\!xB\:$ has integer inverse for all $\rm\:x.$