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1 Let An be the n x n matrix with ai,i = 2 (i=1 to n), ai,i+1 = 1 (i = 1,2,..., n-1), ai+1,i = -1 (i = 1,2,..., n-1) and all other aij = 0. Show by induction on n that √2 det (An) = (1 + √2)¹+¹ − (1 − √2)”+¹] What is da? [Hint: use expansion along row/down column to get an expression for dn in terms of d. and du al

1 Let An be the n x n matrix with ai,i = 2 (i=1 to n), ai,i+1 = 1 (i = 1,2,..., n-1), ai+1,i = -1 (i = 1,2,..., n-1) and all other aij = 0. Show by induction on n that √2 det (An) = (1 + √2)¹+¹ − (1 − √2)”+¹] What is da? [Hint: use expansion along row/down column to get an expression for dn in terms of d. and du al

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.5: Subspaces, Basis, Dimension, And Rank
Problem 63EQ
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1 Let An be the nx n matrix with = ai, i 2 (i=1 to n), ai,i+1 = 1 (i = 1, 2,..., n 1), ai+1,i = -1 (i=1,2,..., n-1) and all other induction on n that aij = 0. Show by det(An) = √²[(1 + √2)”+¹ − (1 − √√2)~+¹] - What is da? [Hint: use expansion along row/down column to get an expression for dn in terms of dn-1 and dn-2.]
Transcribed Image Text:1 Let An be the nx n matrix with = ai, i 2 (i=1 to n), ai,i+1 = 1 (i = 1, 2,..., n 1), ai+1,i = -1 (i=1,2,..., n-1) and all other induction on n that aij = 0. Show by det(An) = √²[(1 + √2)”+¹ − (1 − √√2)~+¹] - What is da? [Hint: use expansion along row/down column to get an expression for dn in terms of dn-1 and dn-2.]
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