# SOLUTION: Could you please help me with this problem: Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1. a) show that D is idempotent b) Show that if X is a n

Algebra ->  Algebra  -> Matrices-and-determiminant -> SOLUTION: Could you please help me with this problem: Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1. a) show that D is idempotent b) Show that if X is a n      Log On

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 Question 376125: Could you please help me with this problem: Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1. a) show that D is idempotent b) Show that if X is a nonsingular matrix and A = XDX^(-1), then A is idempotent.Found 2 solutions by jim_thompson5910, robertb:Answer by jim_thompson5910(28717)   (Show Source): You can put this solution on YOUR website!a) The matrix D is idempotent iff Since is the matrix where each diagonal entry is squared, and and , this means that EVERY diagonal element will NOT change. So each entry of is equal to its corresponding entry of So this means that and that the matrix D is idempotent. Unfortunately I can't type out generalized matrices here, but hopefully you can see it. b) Let's assume that X is nonsingular (ie X is invertible). So this means that exists and So if we want A to be idempotent, then we have to show that So since using the associative property since because from part a) we proved that Using the definition that So since we've shown that , this means that matrix A is idempotent. Hopefully this is clear. If not, let me know. Answer by robertb(4012)   (Show Source): You can put this solution on YOUR website!a) A matrix A is idempotent if . Let D = [d(k,j)] be a diagonal nxn matrix Suppose the diagonal entry d(k,k) = 0. Then the dot product of the kth row of D and the kth column of D (when we're determining ) is also 0 (because the rest of the entries in that row and column are zeroes). If d(k,k) = 1, then the dot product of the kth row of D and the kth column of D is also 1, (again because the rest of the entries in that row and column are zeroes). Thus and D have the same diagonal elements. But the product of any two compatible diagonal matrices is also a diagonal matrix, and thus all the off-diagonal entries are also zero. Therefore . b) , from the condition of (a) above.