Question

Let Aand B bea$\mathbf{2}\mathbf{\times }\mathbf{2}$ matrices with integer entries such that $\mathit{A}\mathbf{,}\mathit{A}\mathbf{+}\mathit{B}\mathbf{,}\mathit{A}\mathbf{+}\mathbf{2}\mathit{B}\mathbf{,}\mathit{A}\mathbf{+}\mathbf{3}\mathit{B}$ , and $\mathit{A}\mathbf{+}\mathbf{4}\mathit{B}$are all invertible matrices whose inverses have integer entries. Show that $\mathit{A}\mathbf{+}\mathbf{5}\mathit{B}$ is invertible and that it’s inverse has integer entries. This question was in the William Lowell Putnam Mathematical Competition in 1994. Hint: Consider the function $\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\mathit{d}\mathit{e}\mathit{t}\mathbf{\right(}\mathit{A}\mathbf{+}\mathit{t}\mathit{B}\mathbf{)}{\mathbf{)}}^{\mathbf{2}}\mathbf{-}\mathbf{1}$. Shows that this is a polynomial; what can you say about its degree? Find the values $\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$, using Exercise 53. Now you can determine $\mathit{f}\left(t\right)$by using a familiar result: If a polynomial $\mathit{f}\left(t\right)$of degree $\mathbf{\le }\mathit{m}$has more than zeros, then$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$ for all t.

Short answer

Therefore, $A+5B$is also an invertible matrix, with integer entries, whose inverse has all integer entries.

Step-by-step solution

Matrix Definition 

Matrix is a setof numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows andn columns, the matrix is said to be a “ mby n” matrix, written “ $m\times n$.”

To find  

Since,$A,A+B,A+2B,A+3B$and $A+4B$are invertible matrices with integer entries whose inverses also have integer entries, then all of their determinants are $\pm 1$.

Consider $f\left(t\right)=\left(det\right(A+tB){)}^2-1$. Since these are all $2\times 2$ matrices, f is a polynomial, whose degree is 4.

We have $f\left(0\right)=f\left(1\right)=f\left(2\right)=f\left(3\right)=f\left(4\right)=0$.

In other words, we have $f\left(t\right)=0$for more values of $t\in \mathrm{ℝ}$than the degree of . Thus, $f\left(t\right)=0,\forall t\in \mathrm{ℝ}$, therefore specifically for $t=5$as well. Therefore,$A+5B$is also an invertible matrix, with integer entries, whose inverse has all integer entries.