Answer: Let J be the n × n matrix of all 1’s, and consider

Chapter , Problem 15E

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Problem 15E

Let J be the n × n matrix of all 1’s, and consider

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?

Reference:

Let J be the n × n matrix of all 1’s, and consider

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.

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.

c. Find the determinant of the resulting matrix from (b).