(from the 1994 Putnam Exam) Let A and B be 2 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. Prove that A + 5B is invertible and that its inverse has integer entries. (Hint: Use Exercise 7.)

7.4 Exponential and Logarithmic Equations Copyright © Cengage Learning. All rights reserved. Introduction There are two basic strategies for solving exponential or logarithmic equations. The first is based on the One-to-One Properties and the second is based on the Inverse Properties. One-to-One Properties a = a if and only if x = y. logax = log a if and only if x = y. Inverse Properties a loa = x log a = x 2 Solving Simple Equations 3 Introduction 4 Example 1 Solve for x. a) 64 3x ▯ 32 2