Explanation Given information: Show the matrix A from Exercise 16. A = [ a b b ⋯ b b a b ⋯ b b b a ⋯ b ⋮ ⋮ ⋮ ⋱ ⋮ b b b ⋯ a ] Show the matrix B and matrix C as follows: B = [ a − b b b ⋯ b 0 a b ⋯ b 0 b a ⋯ b ⋮ ⋮ ⋮ ⋱ ⋮ 0 b b ⋯ a ] C = [ b b b ⋯ b b a b ⋯ b b b a ⋯ b ⋮ ⋮ ⋮ ⋱ ⋮ b b b ⋯ a ] The first column of matrix equals the sum of the first column of matrix B and C . The linearity property of the determinant function is as follows: det A = det B + det C Calculation: Show the matrix B and C for n = 2 as follows: B = [ a − b b 0 a ] C = [ b b b a ] Calculate the determinant of B and C as follows: det B = ( a − b ) a − b × 0 = a 2 − a b det C = b a − b × b = a b − b 2 Calculate the sum of det B and det C as follows: det B + det C = ( a 2 − a b ) + ( a b − b 2 ) = a 2 − a b + a b − b 2 = a 2 − b 2 = ( a − b ) ( a + b ) det B + det C = ( a − b ) [ a + ( 2 − 1 ) b ] = ( a − b ) 2 − 1 [ a + ( 2 − 1 ) b ] Thus, the formula det A = ( a − b ) n − 1 [ a + ( n − 1 ) b ] holds good for n = 2 . Assume the formula det A = ( a − b ) n − 1 [ a + ( n − 1 ) b ] holds good for ( k − 1 ) ( k − 1 ) . Let the matrix A, B, and C be the matrix be k × k

Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

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ISBN: 9781323132098

Author: Thomas, Lay

Publisher: PEARSON C

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Chapter 3, Problem 17SE

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To prove: the formula detA=(a−b)n−1[a+(n−1)b] by mathematical induction.

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Chapter 3, Problem 16SE

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To prove: the formula det A = ( a − b ) n − 1 [ a + ( n − 1 ) b ] by mathematical induction.

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To prove: the formula detA=(a−b)n−1[a+(n−1)b] by mathematical induction.

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Chapter 3 Solutions