Solving simple matrix equations


	Solving simple matrix equations
coburgmaths
9-14-2009 4:51 AM
131 views
coburgmaths says:
Instructions on how to solve simple matrix equations. Some example given and questions to do.
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<div style="margin: 12px 0px; font-family: arial; color: #333333; background: #ffffff; border: solid 4px #e5e5e5; width: 100%; clear: left;"><div class="CM_CTB_Content_Wrap" style="margin: 0px; padding: 0px;background-color: #ffffff;"><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://www.clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content.clipmarks.com/blog_embed/599bee31-5222-4b57-a818-20e3195c9bb2/2528D965-C8B9-42C6-A6FC-0BDA24793D49/" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml" href="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml" style="font-size: 11px;">www.algebralab.org</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml">Now we want to use these matrices to solve our <A href="javascript:def('/Glossary/glossaryterm.aspx?word=System', 500, 500);" class="glossarylink">system</A> of equations. To do so, we will use the calculator, find an inverse, and multiply matrices. If you need help with any of those topics, click on the links below.      </blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><UL type="square"><li style="margin-left:16px;padding-left: 0px;"><A href="http://www.algebralab.org/lessons/lesson.aspx?file=algebra_matrix_intro.xml">Introduction to matrices</A></LI><li style="margin-left:16px;padding-left: 0px;"><A href="http://www.algebralab.org/lessons/lesson.aspx?file=calculator_matrices1.xml">Entering matrices in the calculator</A></LI><li style="margin-left:16px;padding-left: 0px;"><A href="http://www.algebralab.org/lessons/lesson.aspx?file=calculator_matrices2.xml">Inverse matrices on the calculator</A></LI><li style="margin-left:16px;padding-left: 0px;"><A href="http://www.algebralab.org/lessons/lesson.aspx?file=algebra_matrix_operations.xml">Operations with matrices by hand</A></LI><li style="margin-left:16px;padding-left: 0px;"><A href="http://www.algebralab.org/lessons/lesson.aspx?file=calculator_matrices2.xml">Operations with matrices on the calculator</A></LI></UL></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><DIV>         Let’s begin by entering       <IMG width="24" height="20" border="0" xd:inline="" xmlns:xd="http://schemas.microsoft.com/office/infopath/2003" src="http://www.algebralab.org/img/cb8a830e-328a-4e4c-a5ab-68561f0069a8.gif" tabindex="-1" />         and         <IMG width="23" height="20" border="0" xd:inline="" xmlns:xd="http://schemas.microsoft.com/office/infopath/2003" src="http://www.algebralab.org/img/c5546133-57fd-4034-a7a1-61f3eddc6bff.gif" tabindex="-1" />         into the calculator.</DIV></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><table background="undefined" bgcolor=""><tr><TD colspan="2"><SPAN class="redfont" id="lessonctrl_ctl00_ctl00_number"><BR />Examples</SPAN></TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><table background="undefined" bgcolor=""><tr><TD class="interior" colspan="2"><IMG width="95" height="41" border="0" xd:inline="" xmlns:xd="http://schemas.microsoft.com/office/infopath/2003" src="http://www.algebralab.org/img/c202ea46-f918-41fc-80cc-45e1a57de365.gif" tabindex="-1" xmlns="http://www.w3.org/1999/xhtml" /></TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><table background="undefined" bgcolor=""><tr><TD colspan="2"><SPAN class="brownfont">What is your answer?</SPAN>
<INPUT type="text" id="lessonctrl_ctl00_ctl00_ctl01_fib" name="lessonctrl$ctl00$ctl00$ctl01$fib" /> </TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><table background="undefined" bgcolor=""><tr><TD class="interior" colspan="2"><IMG width="69" height="41" border="0" xd:inline="" xmlns:xd="http://schemas.microsoft.com/office/infopath/2003" src="http://www.algebralab.org/img/8a2f4370-0bc9-478e-aeda-cc21c4cc5980.gif" tabindex="-1" xmlns="http://www.w3.org/1999/xhtml" /></TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><table background="undefined" bgcolor=""><tr><TD colspan="2"><SPAN class="brownfont">What is your answer?</SPAN>
<INPUT type="text" id="lessonctrl_ctl00_ctl00_ctl02_fib" name="lessonctrl$ctl00$ctl00$ctl02$fib" /> </TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><table background="undefined" bgcolor=""><tr><TD class="interior" colspan="2"><IMG width="84" height="41" border="0" xd:inline="" xmlns:xd="http://schemas.microsoft.com/office/infopath/2003" src="http://www.algebralab.org/img/f337be9c-57f9-455c-80c6-a7514ea5374d.gif" tabindex="-1" xmlns="http://www.w3.org/1999/xhtml" /></TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><table background="undefined" bgcolor=""><tr><TD colspan="2"><SPAN class="brownfont">What is your answer?</SPAN>
<INPUT type="text" id="lessonctrl_ctl00_ctl00_ctl03_fib" name="lessonctrl$ctl00$ctl00$ctl03$fib" /> </TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_matrix_systems.xml"><TBODY><TR><TD valign="top"><IMG src="http://www.algebralab.org/images/hourglass.jpg" alt="Example" /></TD><TD class="interior" colspan="2"><IMG width="113" height="67" border="0" xd:inline="" xmlns:xd="http://schemas.microsoft.com/office/infopath/2003" src="http://www.algebralab.org/img/8ef3d20e-20ca-4fea-b5b8-360824195fd4.gif" tabindex="-1" xmlns="http://www.w3.org/1999/xhtml" /></TD></TR><TR><TD></TD><TD width="35px" valign="top"><INPUT type="image" align="absmiddle" alt="Hint" src="http://www.algebralab.org/lessons/../images/hint.jpg" id="lessonctrl_ctl00_ctl00_ctl04_hintButton" name="lessonctrl$ctl00$ctl00$ctl04$hintButton" /></TD><TD width="100%"></TD></TR><TR><TD valign="top"></TD><TD colspan="2"><SPAN class="brownfont">What is your answer?</SPAN>
<INPUT type="text" id="lessonctrl_ctl00_ctl00_ctl04_fib" name="lessonctrl$ctl00$ctl00$ctl04$fib" /> </TD></TR><TR><TD><IMG src="http://www.algebralab.org/images/spacer35.gif" /></TD><TD valign="top"></TD><TD class="greenfont"></TD></TR></TBODY></blockquote><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://www.clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content6.clipmarks.com/images/clip-icon.gif" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://www.edhelper.com/Matrices16.htm" href="http://www.edhelper.com/Matrices16.htm" style="font-size: 11px;">www.edhelper.com</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.edhelper.com/Matrices16.htm"><DIV>
<B>Solve the matrix equation.</B></DIV></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.edhelper.com/Matrices16.htm"><TABLE width="100%" border="0"><TBODY><TR valign="top"><TD width="3%">1.  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  0  </TD><TD align="center">  11  </TD></TR><TR><TD align="center">  5  </TD><TD align="center">  8  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD><TD>  X  </TD><TD>  =  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  99  </TD><TD align="center">  -33  </TD></TR><TR><TD align="center">  77  </TD><TD align="center">  -9  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE><BR /><BR /><BR /><BR /><BR /></TD></TR></TBODY></TABLE></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.edhelper.com/Matrices16.htm"><table background="undefined" bgcolor=""><tr><TD width="33%"><TABLE width="100%" border="0"><TBODY><TR valign="top"><TD width="3%">6.  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  2  </TD><TD align="center">  7  </TD></TR><TR><TD align="center">  1  </TD><TD align="center">  5  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD><TD>  X  </TD><TD>  =  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  16  </TD><TD align="center">  -71  </TD></TR><TR><TD align="center">  8  </TD><TD align="center">  -52  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE><BR /><BR /><BR /><BR /><BR /></TD></TR></TBODY></TABLE></TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.edhelper.com/Matrices16.htm"><table background="undefined" bgcolor=""><tr><TD width="33%"><TABLE width="100%" border="0"><TBODY><TR valign="top"><TD width="3%">7.  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  -1  </TD><TD align="center">  -9  </TD></TR><TR><TD align="center">  -12  </TD><TD align="center">  -4  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD><TD>  X  </TD><TD>  =  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  -89  </TD><TD align="center">  -52  </TD></TR><TR><TD align="center">  76  </TD><TD align="center">  0  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE><BR /><BR /><BR /><BR /><BR /></TD></TR></TBODY></TABLE></TD></tr></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.edhelper.com/Matrices16.htm"><TABLE width="100%" border="0"><TBODY><TR valign="top"><TD width="3%">10.  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  7  </TD><TD align="center">  -3  </TD></TR><TR><TD align="center">  5  </TD><TD align="center">  0  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD><TD>  X  </TD><TD>  =  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  70  </TD><TD align="center">  65  </TD></TR><TR><TD align="center">  35  </TD><TD align="center">  40  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE><BR /><BR /><BR /><BR /><BR /></TD></TR></TBODY></TABLE></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.edhelper.com/Matrices16.htm"><table background="undefined" bgcolor=""><TR valign="top"><TD width="3%">14.  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  -9  </TD><TD align="center">  -4  </TD></TR><TR><TD align="center">  -2  </TD><TD align="center">  4  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD><TD>  X  </TD><TD>  =  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  68  </TD><TD align="center">  -35  </TD></TR><TR><TD align="center">  20  </TD><TD align="center">  2  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE><BR /><BR /><BR /><BR /><BR /></TD></TR></table></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.edhelper.com/Matrices16.htm"><table background="undefined" bgcolor=""><TR valign="top"><TD width="3%">16.  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  5  </TD><TD align="center">  -12  </TD></TR><TR><TD align="center">  0  </TD><TD align="center">  -2  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD><TD>  X  </TD><TD>  =  </TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD><IMG src="http://www.edhelperclipart.com/clipart/matlb2.jpg" /></TD><TD><TABLE cellspacing="0" cellpadding="0" border="0"><TBODY><TR><TD align="center">  -38  </TD><TD align="center">  13  </TD></TR><TR><TD align="center">  -8  </TD><TD align="center">  8  </TD></TR></TBODY></TABLE></TD><TD><IMG src="http://www.edhelperclipart.com/clipart/matrb2.jpg" /></TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE><BR /><BR /><BR /><BR /><BR /></TD></TR></table></blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/2528D965-C8B9-42C6-A6FC-0BDA24793D49/blog/" title="blog or email this clip"><img src="http://content9.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>
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