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Let n > 2 and let æ1, x2, ... , Xn+1 € R with x; x1 for all 2 < i < n + 1. Define n × n matrices A and B so that for all 1j Hint: You may use the elementary fact that for all r, y E R and m E N fask 202 Cops (AB)ij = +1 Li+1 – X1 vers z" – ym = (x – y) ►»*-'ym-k k=1 ssessable ssessa Sse

Let n > 2 and let æ1, x2, ... , Xn+1 € R with x; x1 for all 2 < i < n + 1. Define n × n matrices A and B so that for all 1j Hint: You may use the elementary fact that for all r, y E R and m E N fask 202 Cops (AB)ij = +1 Li+1 – X1 vers z" – ym = (x – y) ►»*-'ym-k k=1 ssessable ssessa Sse

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.4: Transistion Matrices And Similarity
Problem 15E
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(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
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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