(a) Let n ≥ 2 and let ₁, 2,...,n+1 ER with x x1 for all "n + 1. 1 ≤i, j≤n we have (A) ij = 2 Ti+1 and j-1 Prove that for all 1 ≤i, j≤ner (c) Hint: You may use the elementary fact that for all : SP eressa ses Mo M and A be defined as above. Prove that tak Si sa Si ta (B)ij Py = 2021 det (M) = (x+1-2₁) det (AB), Fist: You cay find it useful to consider the Laplace on of MAB). where A and B are as defined in Part (a). row operations. You may also find Part (a) useful. by - i+1 Xi+1 - X1 R and m EN "SI (b) Let n ≥ 2 and let 1, 2,. ER with x; # x₁ for all 2 ≤ i ≤n+1. Define the (n+1) × (n+1) matrix M so that for all 1≤i, j≤ n+1 we have (M)ij = ¹. Prove that " = fonal = (x - y)rk-lym-k k=1 m Sabine k=1 essessable Assessa Asses UP nx n matrices A and B so that for all det (M) = ((*:+1 - ₁) ) det (4). i=1 nas take task sk, ver kas Cop that its first column, after first performing some relevant SS (d) Consider the following proposition: For any choice of n ≥ 2 and any choice of x₁,x2,..., n E R, if P denotes the n x n matrix with entries (P) = x¹ for 1 ≤i, j≤n, then det (P) = [₁
(a) Let n ≥ 2 and let ₁, 2,...,n+1 ER with x x1 for all "n + 1. 1 ≤i, j≤n we have (A) ij = 2 Ti+1 and j-1 Prove that for all 1 ≤i, j≤ner (c) Hint: You may use the elementary fact that for all : SP eressa ses Mo M and A be defined as above. Prove that tak Si sa Si ta (B)ij Py = 2021 det (M) = (x+1-2₁) det (AB), Fist: You cay find it useful to consider the Laplace on of MAB). where A and B are as defined in Part (a). row operations. You may also find Part (a) useful. by - i+1 Xi+1 - X1 R and m EN "SI (b) Let n ≥ 2 and let 1, 2,. ER with x; # x₁ for all 2 ≤ i ≤n+1. Define the (n+1) × (n+1) matrix M so that for all 1≤i, j≤ n+1 we have (M)ij = ¹. Prove that " = fonal = (x - y)rk-lym-k k=1 m Sabine k=1 essessable Assessa Asses UP nx n matrices A and B so that for all det (M) = ((*:+1 - ₁) ) det (4). i=1 nas take task sk, ver kas Cop that its first column, after first performing some relevant SS (d) Consider the following proposition: For any choice of n ≥ 2 and any choice of x₁,x2,..., n E R, if P denotes the n x n matrix with entries (P) = x¹ for 1 ≤i, j≤n, then det (P) = [₁
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(a) Let n > 2 and let ¤1, x2,... , Xn+1 € R with xi + xi for all 2 < i < n + 1. Define n × n matrices A and B so that for all
1<i,j <n we have (A)ij = x and
4.
(B)ij =
Sa, i<j
Prove that for all 1 < i, j < n
(AB)ij
Ti+1 - x1
Hint: You may use the elementary fact that for all x, y E R and m e N
m
xm – ym = (x – y) > a
xk-1,m-k
k=1
(b) Let n > 2 and let x1, x2,... , xn+1 € R with x; x1 for all 2 <i <n+1. Define the (n+ 1) × (n +1) matrix M so that for all
1<i,j <n+ 1 we have (M);i = x?-. Prove that
det (M) =
(II (#k+1 – ¤1) ) det(AB),
where A and B are as defined in Part (a).
k=1
Hint: You may find it useful to consider the Laplace expansion of M along its first column, after first performing some relevant
tona
mas
row operations. You may also find Part (a) useful.
ver
(c) Let M and A be defined as above. Prove that
n
det(M) =
(Ti+1 – x1) ) det(A).
i=1
sa
(d) Consider the following proposition: For any choice of n >2 and any choice of x1, x2, ..., xn E R, if P denotes the n x n matrix
with entries (P);j = x for 1 < i, j < n, then det(P) = II1<i<j<n (x; – x;). Prove this proposition by induction on n.
Hint: You may find Part (c) useful.
ssess
Asse
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