document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87107>Question 87107</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0201347741/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> If  A is a square matrix with the property  (A with an exponet of 2) = 0, show that (I - A)inverse = A + I.\r<BR>\n" );
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document.write( "Thank you  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #63089 by </B><A HREF=/tutors/aboutme.mpl?userid=longjonsilver><B>longjonsilver(2297)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=longjonsilver><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/cgi-bin/show-face.mpl?user=longjonsilver><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=63089\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Any matrix multiplied by its inverse is equal to the identity matrix, I by definition. So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%5E%28-1%29%28I-A%29+=+I+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%5E%28-1%29%28I-A%29+=+I+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Post multiply both sides by (I+A):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%5E%28-1%29%28I-A%29%28I%2BA%29+=+I%28I%2BA%29+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%5E%28-1%29%28I-A%29%28I%2BA%29+=+I%28I%2BA%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%28I%2BA%29+=+I%5E2+%2B+IA+-+AI+-+A%5E2+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%28I%2BA%29+=+I%5E2+%2B+IA+-+AI+-+A%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%28I%2BA%29+=+I%5E2+%2B+IA+-+IA+-+A%5E2+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%28I%2BA%29+=+I%5E2+%2B+IA+-+IA+-+A%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%28I%2BA%29+=+I+%2B+IA+-+IA+-+A%5E2+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%28I%2BA%29+=+I+%2B+IA+-+IA+-+A%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%28I%2BA%29+=+I+-+A%5E2+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%28I%2BA%29+=+I+-+A%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%28I%2BA%29+=+I+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%28I%2BA%29+=+I+\"   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=+%28I-A%29%5E%28-1%29%28I-A%29%28I%2BA%29+=+I%28I%2BA%29+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%5E%28-1%29%28I-A%29%28I%2BA%29+=+I%28I%2BA%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%5E%28-1%29+=+I%28I%2BA%29+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%5E%28-1%29+=+I%28I%2BA%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28I-A%29%5E%28-1%29+=+%28I%2BA%29+ ALIGN=MIDDLE  ALT=\"+%28I-A%29%5E%28-1%29+=+%28I%2BA%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "cheers<BR>\n" );
document.write( "Jon<BR>\n" );
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