document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=376125>Question 376125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could you please help me with this problem:\r<BR>\n" );
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document.write( "Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1.\r<BR>\n" );
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document.write( "a) show that D is idempotent<BR>\n" );
document.write( "b) Show that if X is a nonsingular matrix and A = XDX^(-1), then A is idempotent. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #267521 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=267521\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) The matrix D is idempotent iff <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=D ALIGN=MIDDLE  ALT=\"D%5E2=D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2 ALIGN=MIDDLE  ALT=\"D%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the matrix where each diagonal entry is squared, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%5E2=0 ALIGN=MIDDLE  ALT=\"0%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E2=1 ALIGN=MIDDLE  ALT=\"1%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that EVERY diagonal element will NOT change. So each entry of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2 ALIGN=MIDDLE  ALT=\"D%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to its corresponding entry of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=D ALIGN=MIDDLE  ALT=\"D%5E2=D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that the matrix D is idempotent.\r<BR>\n" );
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document.write( "Unfortunately I can't type out generalized matrices here, but hopefully you can see it.\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Let's assume that X is nonsingular (ie X is invertible). So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E%28-1%29 ALIGN=MIDDLE  ALT=\"X%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > exists and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%2AX%5E%28-1%29=I ALIGN=MIDDLE  ALT=\"X%2AX%5E%28-1%29=I\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So if we want A to be idempotent, then we have to show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=A ALIGN=MIDDLE  ALT=\"A%5E2=A\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=%28XDX%5E%28-1%29%29%5E2 ALIGN=MIDDLE  ALT=\"A%5E2=%28XDX%5E%28-1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=%28XDX%5E%28-1%29%29%28XDX%5E%28-1%29%29 ALIGN=MIDDLE  ALT=\"A%5E2=%28XDX%5E%28-1%29%29%28XDX%5E%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2=X%2AX ALIGN=MIDDLE  ALT=\"X%5E2=X%2AX\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=XD%28X%5E%28-1%29X%29DX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"A%5E2=XD%28X%5E%28-1%29X%29DX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the associative property\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=XD%28I%29DX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"A%5E2=XD%28I%29DX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%2AX%5E%28-1%29=I ALIGN=MIDDLE  ALT=\"X%2AX%5E%28-1%29=I\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=XD%2ADX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"A%5E2=XD%2ADX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=XD%5E2X%5E%28-1%29 ALIGN=MIDDLE  ALT=\"A%5E2=XD%5E2X%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=XDX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"A%5E2=XDX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > because from part a) we proved that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=D ALIGN=MIDDLE  ALT=\"D%5E2=D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=A ALIGN=MIDDLE  ALT=\"A%5E2=A\"   DISABLED_style= \"max-width:90%; height:auto;\" > Using the definition that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=XDX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"A=XDX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So since we've shown that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2=A ALIGN=MIDDLE  ALT=\"A%5E2=A\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that matrix A is idempotent.\r<BR>\n" );
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document.write( "Hopefully this is clear. If not, let me know.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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