document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=28312>Question 28312</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ok - I am having problems with solving the follwoing equation -- can you please help me out. Please note that the M amounts within the brackets are in sets of four -- 2 on top of 2 - for example M= a 0(being on the top line while 0  b are right below it -- lining up the a and the 0 and then the 0 and the b.\r<BR>\n" );
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document.write( "Let M= [a 0] <BR>\n" );
document.write( "       [0 b] with a, b real numbers.\r<BR>\n" );
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document.write( "a) Solve that M(squared) =MM= [a(squared) 0]<BR>\n" );
document.write( "                              [0        b(squared)].\r<BR>\n" );
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document.write( "b) What is det(M)?\r<BR>\n" );
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document.write( "c) Under what condition(s) does M(inverse) exist?\r<BR>\n" );
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document.write( "d) What is M(inverse)?\r<BR>\n" );
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document.write( "e) Suppose M= [2  0]<BR>\n" );
document.write( "              [0  1]. What M(tenth power)? (hint: find M(squared), M(cube).... is there apattern? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15403 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=15403\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let M= [a 0]<BR>\n" );
document.write( ".......[0 b] with a, b real numbers.<BR>\n" );
document.write( "a) Solve that M(squared) =MM= [a(squared) 0]<BR>\n" );
document.write( "..............................[0 b(squared)].<BR>\n" );
document.write( "MM=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2Ca%2C0%2C0%2Cb%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2Ca%2C0%2C0%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2Ca%2C0%2C0%2Cb%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2Ca%2C0%2C0%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2Ca%2Aa%2B0%2A0%2Ca%2A0%2B0%2Ab%2C0%2Aa%2Bb%2A0%2C0%2A0%2Bb%2Ab%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2Ca%2Aa%2B0%2A0%2Ca%2A0%2B0%2Ab%2C0%2Aa%2Bb%2A0%2C0%2A0%2Bb%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2Ca%5E2%2C0%2C0%2Cb%5E2%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2Ca%5E2%2C0%2C0%2Cb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b) What is det(M)?<BR>\n" );
document.write( "=a*b-0*)=ab<BR>\n" );
document.write( "c) Under what condition(s) does M(inverse) exist?<BR>\n" );
document.write( "det(m)=ab should not equal zero...that is either a or b or both should not be zero <BR>\n" );
document.write( "d) What is M(inverse)?<BR>\n" );
document.write( "step1..replace each element with their minor<BR>\n" );
document.write( "minor matrix=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2Cb%2C0%2C0%2Ca%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2Cb%2C0%2C0%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "step2....multiply each element with (-1)^(m+n),where m and n are row and column numbers.<BR>\n" );
document.write( "cofactor matrix=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C%2Bb%2C-0%2C-0%2C%2Ba%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C%2Bb%2C-0%2C-0%2C%2Ba%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "step3......transpose the above<BR>\n" );
document.write( "transpose =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2Cb%2C0%2C0%2Ca%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2Cb%2C0%2C0%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "step 4...divide by determinant to get inverse<BR>\n" );
document.write( "inverse of M = (1/ab)*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2Cb%2C0%2C0%2Ca%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2Cb%2C0%2C0%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "e) Suppose M= [2 0]<BR>\n" );
document.write( "..............[0 1]. What M(tenth power)? (hint: find M(squared), M(cube).... is there apattern?<BR>\n" );
document.write( "USING ABOVE WE GET<BR>\n" );
document.write( "MM=M^2=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C4%2C0%2C0%2C1%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C4%2C0%2C0%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "MMM=M^3=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C4%2C0%2C0%2C1%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C4%2C0%2C0%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C2%2C0%2C0%2C1%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C2%2C0%2C0%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C8%2C0%2C0%2C1%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C8%2C0%2C0%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "YES THERE IS A PATTERN <BR>\n" );
document.write( "M=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C2%2C0%2C0%2C1%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C2%2C0%2C0%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "M^2=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C2%5E2%2C0%2C0%2C1%5E2%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C2%5E2%2C0%2C0%2C1%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "M^3=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C2%5E3%2C0%2C0%2C1%5E3%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C2%5E3%2C0%2C0%2C1%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "...............................HENCE<BR>\n" );
document.write( "M^10=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C2%5E10%2C0%2C0%2C1%5E10%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C2%5E10%2C0%2C0%2C1%5E10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );