document.write( "Question 376125: Could you please help me with this problem:\r
\n" ); document.write( "\n" ); document.write( "Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1.\r
\n" ); document.write( "\n" ); document.write( "a) show that D is idempotent
\n" ); document.write( "b) Show that if X is a nonsingular matrix and A = XDX^(-1), then A is idempotent. \n" ); document.write( "

Algebra.Com's Answer #267522 by robertb(5830) $\"\"$ $\"About$

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a) A matrix A is idempotent if $\"A%5E2+=+A\"$ . 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 $\"D%5E2\"$ ) is also 0 (because the rest of the entries in that row and column are zeroes).
\n" ); document.write( "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 $\"D%5E2\"$ 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 $\"D%5E2+=+D\"$ .\r
\n" ); document.write( "\n" ); document.write( "b)

, from the condition of (a) above. \n" ); document.write( "

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