Chapter 3, Problem 30SE

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

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

Confirm that det A = (a− b)ⁿ⁻¹ [a + (n − 1 )b] as follows:

1. a. Subtract row 2 from row 1, row 3 from row 2, and so on, and explain why this does not change the determinant of the matrix.
2. b. With the resulting matrix from part (a), add column 1 to column 2, then add this new column 2 to column 3, and so on, and explain why this does not change the determinant.
3. c. Find the determinant of the resulting matrix from (b).