document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251271>Question 251271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a+b+c+d+e+f=0<BR>\n" );
document.write( "a-2b=0<BR>\n" );
document.write( "a+2b+3d+2e=220<BR>\n" );
document.write( "1+3c+2d+2f=110<BR>\n" );
document.write( "a+2b+4d+e+5f=240<BR>\n" );
document.write( "a+2d+3e+4f=165\r<BR>\n" );
document.write( "\n" );
document.write( "*write the equations in matrix form AX=B.\r<BR>\n" );
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document.write( "*using an appropriate software package find the inverse of A.\r<BR>\n" );
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document.write( "*solve the above equations using the Matrix Inverse Method.\r<BR>\n" );
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document.write( "*check your answer using Cramer's rule and the Gaussian Elimination Method. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184464 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20054)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184464\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><font size = 4 color = \"indigo\"><b>\r\n" );
document.write( "I think you must have mistyped the equation \r\n" );
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document.write( "1+3c+2d+2f=110\r\n" );
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document.write( "as none of the others had a number in front\r\n" );
document.write( "that you had to subtract from both sides, so\r\n" );
document.write( "if the answer I get is not the right one it's\r\n" );
document.write( "because you mistyped that 1 in front of the \r\n" );
document.write( "fourth equation\r\n" );
document.write( "\r\n" );
document.write( "a+b+c+d+e+f=0\r\n" );
document.write( "a-2b=0\r\n" );
document.write( "a+2b+3d+2e=220\r\n" );
document.write( "1+3c+2d+2f=110\r\n" );
document.write( "a+2b+4d+e+5f=240\r\n" );
document.write( "a+2d+3e+4f=165 \r\n" );
document.write( "\r\n" );
document.write( "1a + 1b + 1c + 1d + 1e + 1f =   0\r\n" );
document.write( "1a - 2b + 0c + 0d + 0e + 0f =   0\r\n" );
document.write( "1a + 2b + 0c + 3d + 2e + 0f = 220\r\n" );
document.write( "0a + 0b + 3c + 2d + 0e + 2f = 110-1\r\n" );
document.write( "1a + 2b + 0c + 4d + 1e + 5f = 240\r\n" );
document.write( "1a + 0b + 0c + 2d + 3e + 4f = 165\r\n" );
document.write( "\r\n" );
document.write( "Simplifying the equation that I think\r\n" );
document.write( "you mistyped:\r\n" );
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document.write( "1a + 1b + 1c + 1d + 1e + 1f =   0\r\n" );
document.write( "1a - 2b + 0c + 0d + 0e + 0f =   0\r\n" );
document.write( "1a + 2b + 0c + 3d + 2e + 0f = 220\r\n" );
document.write( "0a + 0b + 3c + 2d + 0e + 2f = 109\r\n" );
document.write( "1a + 2b + 0c + 4d + 1e + 5f = 240\r\n" );
document.write( "1a + 0b + 0c + 2d + 3e + 4f = 165 \r\n" );
document.write( "\r\n" );
document.write( "This is the matrix AX=B form:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%286%2C6%2C%0D%0A1%2C++1%2C++1%2C++1%2C++1%2C++1%2C+%0D%0A1%2C+-2%2C++0%2C++0%2C++0%2C++0%2C+%0D%0A1%2C++2%2C++0%2C++3%2C++2%2C++0%2C+%0D%0A0%2C++0%2C++3%2C++2%2C++0%2C++2%2C+%0D%0A1%2C++2%2C++0%2C++4%2C++1%2C++5%2C+%0D%0A1%2C++0%2C++0%2C++2%2C++3%2C++4%29%29%2A%28matrix%286%2C1%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%29%29=%28matrix%286%2C1%2C0%2C0%2C220%2C109%2C240%2C165%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r\n" );
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document.write( "The inverse of that matrix is:\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%286%2C6%2C%0D%0A270%2F341%2C++++13%2F32%2C++-38%2F341%2C+-90%2F341%2C+46%2F341%2C+-80%2F341%2C%0D%0A135%2F341%2C++++-9%2F31%2C++-19%2F341%2C+-45%2F341%2C+23%2F341%2C+-40%2F341%2C%0D%0A92%2F341%2C+++++-2%2F31%2C++-18%2F341%2C++83%2F341%2C-50%2F341%2C+-2%2F341%2C%0D%0A-168%2F341%2C++++5%2F31%2C++107%2F341%2C++56%2F341%2C+32%2F341%2C+-26%2F341%2C%0D%0A-18%2F341%2C++++-5%2F31%2C+++48%2F341%2C+++6%2F341%2C+-94%2F341%2C+119%2F341%2C%0D%0A30%2F341%2C+++++-2%2F31%2C++-80%2F341%2C+-10%2F341%2C+43%2F341%2C++29%2F341%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "\r\n" );
document.write( "We multiply both sides of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AX=B ALIGN=MIDDLE  ALT=\"AX=B\"   DISABLED_style= \"max-width:90%; height:auto;\" > equation by that\r\n" );
document.write( "inverse matrix, getting it in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E%28-1%29%28AX%29=A%5E%28-1%29B ALIGN=MIDDLE  ALT=\"A%5E%28-1%29%28AX%29=A%5E%28-1%29B\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%286%2C6%2C%0D%0A270%2F341%2C++++13%2F32%2C++-38%2F341%2C+-90%2F341%2C+46%2F341%2C+-80%2F341%2C%0D%0A135%2F341%2C++++-9%2F31%2C++-19%2F341%2C+-45%2F341%2C+23%2F341%2C+-40%2F341%2C%0D%0A92%2F341%2C+++++-2%2F31%2C++-18%2F341%2C++83%2F341%2C-50%2F341%2C+-2%2F341%2C%0D%0A-168%2F341%2C++++5%2F31%2C++107%2F341%2C++56%2F341%2C+32%2F341%2C+-26%2F341%2C%0D%0A-18%2F341%2C++++-5%2F31%2C+++48%2F341%2C+++6%2F341%2C+-94%2F341%2C+119%2F341%2C%0D%0A30%2F341%2C+++++-2%2F31%2C++-80%2F341%2C+-10%2F341%2C+43%2F341%2C++29%2F341%29%29%2A%0D%0A%0D%0A%28%28matrix%286%2C6%2C%0D%0A1%2C++1%2C++1%2C++1%2C++1%2C++1%2C+%0D%0A1%2C+-2%2C++0%2C++0%2C++0%2C++0%2C+%0D%0A1%2C++2%2C++0%2C++3%2C++2%2C++0%2C+%0D%0A0%2C++0%2C++3%2C++2%2C++0%2C++2%2C+%0D%0A1%2C++2%2C++0%2C++4%2C++1%2C++5%2C+%0D%0A1%2C++0%2C++0%2C++2%2C++3%2C++4%29%29%2A%28matrix%286%2C1%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%29%29%29=%0D%0A%0D%0A%0D%0A%28matrix%286%2C6%2C%0D%0A270%2F341%2C++++13%2F32%2C++-38%2F341%2C+-90%2F341%2C+46%2F341%2C+-80%2F341%2C%0D%0A135%2F341%2C++++-9%2F31%2C++-19%2F341%2C+-45%2F341%2C+23%2F341%2C+-40%2F341%2C%0D%0A92%2F341%2C+++++-2%2F31%2C++-18%2F341%2C++83%2F341%2C-50%2F341%2C+-2%2F341%2C%0D%0A-168%2F341%2C++++5%2F31%2C++107%2F341%2C++56%2F341%2C+32%2F341%2C+-26%2F341%2C%0D%0A-18%2F341%2C++++-5%2F31%2C+++48%2F341%2C+++6%2F341%2C+-94%2F341%2C+119%2F341%2C%0D%0A30%2F341%2C+++++-2%2F31%2C++-80%2F341%2C+-10%2F341%2C+43%2F341%2C++29%2F341%29%29%2A%28matrix%286%2C1%2C0%2C0%2C220%2C109%2C240%2C165%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r\n" );
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document.write( "Matrix multiplication. though not commutative, is\r\n" );
document.write( "associative, so we move the parentheses and change the\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E%28-1%29%28AX%29=A%5E%28-1%29B ALIGN=MIDDLE  ALT=\"A%5E%28-1%29%28AX%29=A%5E%28-1%29B\"   DISABLED_style= \"max-width:90%; height:auto;\" > form to the\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%5E%28-1%29A%29X=A%5E%28-1%29B ALIGN=MIDDLE  ALT=\"%28A%5E%28-1%29A%29X=A%5E%28-1%29B\"   DISABLED_style= \"max-width:90%; height:auto;\" > form\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28matrix%286%2C6%2C%0D%0A270%2F341%2C++++13%2F32%2C++-38%2F341%2C+-90%2F341%2C+46%2F341%2C+-80%2F341%2C%0D%0A135%2F341%2C++++-9%2F31%2C++-19%2F341%2C+-45%2F341%2C+23%2F341%2C+-40%2F341%2C%0D%0A92%2F341%2C+++++-2%2F31%2C++-18%2F341%2C++83%2F341%2C-50%2F341%2C+-2%2F341%2C%0D%0A-168%2F341%2C++++5%2F31%2C++107%2F341%2C++56%2F341%2C+32%2F341%2C+-26%2F341%2C%0D%0A-18%2F341%2C++++-5%2F31%2C+++48%2F341%2C+++6%2F341%2C+-94%2F341%2C+119%2F341%2C%0D%0A30%2F341%2C+++++-2%2F31%2C++-80%2F341%2C+-10%2F341%2C+43%2F341%2C++29%2F341%29%29%2A%0D%0A%0D%0A%28matrix%286%2C6%2C%0D%0A1%2C++1%2C++1%2C++1%2C++1%2C++1%2C+%0D%0A1%2C+-2%2C++0%2C++0%2C++0%2C++0%2C+%0D%0A1%2C++2%2C++0%2C++3%2C++2%2C++0%2C+%0D%0A0%2C++0%2C++3%2C++2%2C++0%2C++2%2C+%0D%0A1%2C++2%2C++0%2C++4%2C++1%2C++5%2C+%0D%0A1%2C++0%2C++0%2C++2%2C++3%2C++4%29%29%29%2A%28matrix%286%2C1%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%29%29=%0D%0A%0D%0A%0D%0A%28matrix%286%2C6%2C%0D%0A270%2F341%2C++++13%2F32%2C++-38%2F341%2C+-90%2F341%2C+46%2F341%2C+-80%2F341%2C%0D%0A135%2F341%2C++++-9%2F31%2C++-19%2F341%2C+-45%2F341%2C+23%2F341%2C+-40%2F341%2C%0D%0A92%2F341%2C+++++-2%2F31%2C++-18%2F341%2C++83%2F341%2C-50%2F341%2C+-2%2F341%2C%0D%0A-168%2F341%2C++++5%2F31%2C++107%2F341%2C++56%2F341%2C+32%2F341%2C+-26%2F341%2C%0D%0A-18%2F341%2C++++-5%2F31%2C+++48%2F341%2C+++6%2F341%2C+-94%2F341%2C+119%2F341%2C%0D%0A30%2F341%2C+++++-2%2F31%2C++-80%2F341%2C+-10%2F341%2C+43%2F341%2C++29%2F341%29%29%2A%28matrix%286%2C1%2C0%2C0%2C220%2C109%2C240%2C165%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "\r\n" );
document.write( "Next we do the matrix multiplication on both sides, and\r\n" );
document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E%28-1%29%2AA=I ALIGN=MIDDLE  ALT=\"A%5E%28-1%29%2AA=I\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=IX=A%5E%28-1%29B ALIGN=MIDDLE  ALT=\"IX=A%5E%28-1%29B\"   DISABLED_style= \"max-width:90%; height:auto;\" > form:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A%0D%0A%28matrix%286%2C6%2C%0D%0A1%2C++0%2C++0%2C++0%2C++0%2C++0%2C+%0D%0A0%2C++1%2C++0%2C++0%2C++0%2C++0%2C+%0D%0A0%2C++0%2C++1%2C++0%2C++0%2C++0%2C+%0D%0A0%2C++0%2C++0%2C++1%2C++0%2C++0%2C+%0D%0A0%2C++0%2C++0%2C++0%2C++1%2C++0%2C+%0D%0A0%2C++0%2C++0%2C++0%2C++0%2C++1%29%29%2A%28matrix%286%2C1%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%29%29=%0D%0A%0D%0A%28matrix%286%2C1%2C%0D%0A-20330%2F341%2C+-10165%2F341%2C+-7243%2F341%2C+33034%2F341%2C+8289%2F341%2C-3585%2F341%29%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "When we make the multiplication on the left, we\r\n" );
document.write( "have the final solution, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X=A%5E%28-1%29B ALIGN=MIDDLE  ALT=\"X=A%5E%28-1%29B\"   DISABLED_style= \"max-width:90%; height:auto;\" > form:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%286%2C1%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%29%29=%0D%0A%0D%0A%28matrix%286%2C1%2C%0D%0A-20330%2F341%2C+-10165%2F341%2C+-7243%2F341%2C+33034%2F341%2C+8289%2F341%2C-3585%2F341%29%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "\r\n" );
document.write( "The answer you wanted is probably all different numbers since you \r\n" );
document.write( "probably mistyped that 1 in front of the 4th equation instead of\r\n" );
document.write( "what it should have been.\r\n" );
document.write( "\r\n" );
document.write( "I did the above on a TI-84 calculator.\r\n" );
document.write( "\r\n" );
document.write( "Edwin</PRE>  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );