Chapter 5, Problem 15SE

Let J be the n × n matrix of all 1’s, and consider A = (a − b)I + bJ; that is,

A = $\left[\begin{array}{ccccc}\text{a}& \text{b}& \text{b}& \cdots & \text{b}\\ \text{b}& \text{a}& \text{b}& \cdots & \text{b}\\ \text{b}& \text{b}& \text{a}& \cdots & \text{b}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \text{b}& \text{b}& \text{b}& \cdots & \text{a}\end{array}\right]$

Use the results of Exercise 16 in the Supplementary Exercises for Chapter 3 to show that the eigenvalues of A are a − b and a + (n − 1)b. What are the multiplicities of these eigenvalues?