Let J be the n × n matrix of all 1’s, and consider A = (a - b)I + b J; that is,
A =
Confirm that det A = (a− b)n−1 [a + (n − 1 )b] as follows:
- 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).
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