document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=904029>Question 904029</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let A be an idempotent matrix and X a nonsingular matrix. Show that C= XAX^&#8722;1 is an idempotent matrix. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #548482 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=548482\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We want to show that C is an idempotent matrix\r<BR>\n" );
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document.write( "That means we want to show that C*C = C is true.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+X%2AA%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C+=+X%2AA%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+%28X%2AA%2AX%5E%28-1%29%29%2AC ALIGN=MIDDLE  ALT=\"C%2AC+=+%28X%2AA%2AX%5E%28-1%29%29%2AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > Right multiply both sides by C (see note below)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+%28X%2AA%2AX%5E%28-1%29%29%2A%28X%2AA%2AX%5E%28-1%29%29 ALIGN=MIDDLE  ALT=\"C%2AC+=+%28X%2AA%2AX%5E%28-1%29%29%2A%28X%2AA%2AX%5E%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+X%2AA%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C+=+X%2AA%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+X%2AA%2A%28X%5E%28-1%29%2AX%29%2AA%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C%2AC+=+X%2AA%2A%28X%5E%28-1%29%2AX%29%2AA%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the associative property of matrix multiplication.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+X%2AA%2A%28I%29%2AA%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C%2AC+=+X%2AA%2A%28I%29%2AA%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > The expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E%28-1%29%2AX ALIGN=MIDDLE  ALT=\"X%5E%28-1%29%2AX\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I ALIGN=MIDDLE  ALT=\"I\"   DISABLED_style= \"max-width:90%; height:auto;\" > since X is nonsingular (ie X is invertible)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+X%2A%28A%2AI%29%2AA%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C%2AC+=+X%2A%28A%2AI%29%2AA%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the associative property of matrix multiplication.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+X%2AA%2AA%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C%2AC+=+X%2AA%2AA%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Matrix Multiplicative Identity Property \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+X%2A%28A%2AA%29%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C%2AC+=+X%2A%28A%2AA%29%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the associative property of matrix multiplication.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+X%2AA%2AX%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C%2AC+=+X%2AA%2AX%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Matrix A is idempotent, so A*A = A\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+C ALIGN=MIDDLE  ALT=\"C%2AC+=+C\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace the right side with what it really is\r<BR>\n" );
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document.write( "Because we've shown that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2AC+=+C ALIGN=MIDDLE  ALT=\"C%2AC+=+C\"   DISABLED_style= \"max-width:90%; height:auto;\" > is true, this proves that C is an idempotent matrix.\r<BR>\n" );
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document.write( "Note: You can also use left multiplication to get to the same result (in a very slightly different way, nothing too major though). Left multiplication is a bit different than right multiplication because matrix multiplication is NOT commutative. The proof is effectively the same as shown above which is why I don't need to show it, but it's important to keep this in mind.<BR>\n" );
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