document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=361049>Question 361049</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have worked out Eigenvalues of the following matrix\r<BR>\n" );
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document.write( "1 -1 2<BR>\n" );
document.write( "0 2 0<BR>\n" );
document.write( "3 -3 2\r<BR>\n" );
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document.write( "They are = -1, 2, 4\r<BR>\n" );
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document.write( "I cant seem to work out the eigenvector for -1. Could someone show me step by step how this achieved.  I can then use this to work out 2, and 4.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #257580 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=257580\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For &#955; = -1, replacing -1 into the homogeneous system A - &#955;I = O, we get the augmented homogeneous system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C3%2C0%2C0%2C3%2C-3%2C3%2C0%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C3%2C0%2C0%2C3%2C-3%2C3%2C0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C1%2C0%2C0%2C1%2C-1%2C1%2C0%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C1%2C0%2C0%2C1%2C-1%2C1%2C0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Divide r2 and r3 by 3.<BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C+2%2C-1%2C2%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add row2 to r3<BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C+0%2C-1%2C0%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C+0%2C-1%2C0%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-2*r3 + r1<BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C+0%2C0%2C0%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C+0%2C0%2C0%2C0%2C0%2C1%2C0%2C0%2C1%2C0%2C1%2C0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add r2 to r1<BR>\n" );
document.write( "~<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C+1%2C0%2C1%2C0%2C0%2C1%2C0%2C0%2C0%2C0%2C0%2C0%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C+1%2C0%2C1%2C0%2C0%2C1%2C0%2C0%2C0%2C0%2C0%2C0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Interchange r1 and r3.<BR>\n" );
document.write( "Thus y = 0 and x + z = 0, or z = -x. hence,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C1%2Cx%2Cy%2Cz%29%29=%28matrix%283%2C1%2Cx%2C0%2C-x%29%29+=+x%28matrix%283%2C1%2C1%2C0%2C-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.  The basis for eigenspace for &#955; = -1 is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28matrix%283%2C1%2C1%2C0%2C-1%29%29+ ALIGN=MIDDLE  ALT=\"+%28matrix%283%2C1%2C1%2C0%2C-1%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and this is its eigenvector. </SPAN>\n" );
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