Question

Question: Let A and B be 2 x 2 matrices with integer entries such that A,A+B,A+2B,A+3B, and A+4B are all invertible matrices whose inverses have integer entries. Show that A+5B 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 F(t)=(det(A+tB))² -1. Shows that this is a polynomial; what can you say about its degree? Find the values f(0), f(1), f(2), f(3), f(4),using Exercise 53. Now you can determine f(t) by using a familiar result: If a polynomial f(t) of degree <m has more than m zeros, then f(t)=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 aset of numbers arranged inrowsandcolumnsso as to form a rectangulararray.

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

If there aremrows andncolumns, the matrix is said to be a “mbyn” matrix, written “m×n.”

To find  f(t)

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

Consider F(t)=(det(A+tB))² -1. Since these are all 2×2 matrices, f is a polynomial, whose degree is 4.

We have f(0)=f(1)=f(2)=f(3)=f(4) =0