Let J„ be the n × n matrix whose entries are all equal to 1, and let D(^1, ... , An) be the n × n diagonal matrix whose non-zero entries are 11, ... , An ɛ R. Let x = (x1, .. , Xn) be a row vector in R" and let x' be its transpose, a column vector. 1. Show that xJ„x' > 0, for all x1, . , xn E R. 2. Assume that the scalars 11,... , An are all strictly positive. Show that x D(21, ... , An)x' is strictly positive for all (x1, Xn), unless x1 = … Xn = 0. .... 3. Use questions 2 and 1 above to show that D(^1, ... , An) + AJ, is an invertible matrix when 11, .. , an and 1 are all strictly positive.
Let J„ be the n × n matrix whose entries are all equal to 1, and let D(^1, ... , An) be the n × n diagonal matrix whose non-zero entries are 11, ... , An ɛ R. Let x = (x1, .. , Xn) be a row vector in R" and let x' be its transpose, a column vector. 1. Show that xJ„x' > 0, for all x1, . , xn E R. 2. Assume that the scalars 11,... , An are all strictly positive. Show that x D(21, ... , An)x' is strictly positive for all (x1, Xn), unless x1 = … Xn = 0. .... 3. Use questions 2 and 1 above to show that D(^1, ... , An) + AJ, is an invertible matrix when 11, .. , an and 1 are all strictly positive.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
Problem 85E: Determine if the statement is true or false. If the statement is false, then correct it and make it...
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