document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=176635>Question 176635</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following systems using inverse matrices. can you also show me the steps.. thanks\r<BR>\n" );
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document.write( "2x - y = 2<BR>\n" );
document.write( "2x - 2y = 4\r<BR>\n" );
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document.write( "2x + 3y + 4z = 3<BR>\n" );
document.write( "-13x + 5y - 2z = 3<BR>\n" );
document.write( "-3x + 4y + 3z = 6<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 #131732 by </B><A HREF=/tutors/aboutme.mpl?userid=Mathtut><B>Mathtut(3670)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Mathtut><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=131732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 2x - y = 2<BR>\n" );
document.write( "2x - 2y = 4<BR>\n" );
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document.write( "matrix A =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C2%2C-1%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C2%2C-1%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "matrix C =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C2%2C4%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C2%2C4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "adj A=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C-2%2C1%2C-2%2C2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C-2%2C1%2C-2%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the adjunct of a 2x2 matrix is the matrix in which the main diagonal elements are switched and the other elements are multiplied by -1.<BR>\n" );
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document.write( "the inverse of A or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E-1 ALIGN=MIDDLE  ALT=\"A%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <BR>\n" );
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document.write( "so det A=-2(2)-2(-1)=-2--->-1/2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E-1 ALIGN=MIDDLE  ALT=\"A%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >=(1/det A)(adj A)=-1/2<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C-2%2C1%2C-2%2C2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C-2%2C1%2C-2%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2C-1%2F2%2C1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C1%2C-1%2F2%2C1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "now finally the answer is found by multiplying the inverse of A by matrix C<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E-1 ALIGN=MIDDLE  ALT=\"A%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >mat C<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2C-1%2F2%2C1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C1%2C-1%2F2%2C1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C2%2C4%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C2%2C4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C0%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C0%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "so x=0 and y=-2<BR>\n" );
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document.write( "2x + 3y + 4z = 3<BR>\n" );
document.write( "-13x + 5y - 2z = 3<BR>\n" );
document.write( "-3x + 4y + 3z = 6<BR>\n" );
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document.write( "RHS is the numbers on the right hand side of the equation.<BR>\n" );
document.write( "C1 is column 1...etc<BR>\n" );
document.write( "R1 is row 1...etc\r<BR>\n" );
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document.write( ":Matrix A <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C2%2C3%2C4%2C-13%2C5%2C-2%2C-3%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C2%2C3%2C4%2C-13%2C5%2C-2%2C-3%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "det A is the determinant of A<BR>\n" );
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document.write( "det A=<b>(2)</b> det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2C-2%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2C-2%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-<b>(-13)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C4%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C4%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<b>(-3)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C4%2C5%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C4%2C5%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=46-91+78=33<BR>\n" );
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document.write( "det A=13<BR>\n" );
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document.write( "matrix A(C1 replaced with RHS) lets call  A1<BR>\n" );
document.write( "A1=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C3%2C3%2C4%2C3%2C5%2C-2%2C6%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C3%2C3%2C4%2C3%2C5%2C-2%2C6%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "det A1=b>(3)</b> det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2C-2%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2C-2%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-<b>(3)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C4%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C4%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<b>(6)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C4%2C5%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C4%2C5%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=69+21-156=-66<BR>\n" );
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document.write( "matrix A(C2 replaced with RHS) lets call  A2<BR>\n" );
document.write( "A2=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C2%2C3%2C4%2C-13%2C3%2C-2%2C-3%2C6%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C2%2C3%2C4%2C-13%2C3%2C-2%2C-3%2C6%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "det A2=b>(2)</b> det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C-2%2C6%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C-2%2C6%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-<b>(-13)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C4%2C6%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C4%2C6%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<b>(-3)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C4%2C3%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C4%2C3%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=42-195+54=-99<BR>\n" );
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document.write( "matrix A(C3 replaced with RHS) lets call  A3<BR>\n" );
document.write( "A3=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C2%2C3%2C3%2C-13%2C5%2C3%2C-3%2C4%2C6%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C2%2C3%2C3%2C-13%2C5%2C3%2C-3%2C4%2C6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "det A3=b>(2)</b> det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2C3%2C4%2C6%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2C3%2C4%2C6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-<b>(-13)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C3%2C4%2C6%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C3%2C4%2C6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<b>(-3)</b>det<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C3%2C3%2C5%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C3%2C3%2C5%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=36+78+18=132<BR>\n" );
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