document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30537>Question 30537</A>: <I><SPAN STYLE=\"word-break: break-all;\"> when a question asks you to show something algebraically, DO NOT substitute numbers for the variables and give me an example.  On the other hand, when a question asks you to show something numerically or to provide an example, you may substitute numbers for the variables. Show all work\r<BR>\n" );
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document.write( "Let A and B be two 2x2 matrices.  Show by example that the determinant function is multiplicative, that is,\r<BR>\n" );
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document.write( "det(A)det(B) = det(AB)\r<BR>\n" );
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document.write( "(here you can pick numeric examples for A and B).  \r<BR>\n" );
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document.write( "Bonus: show algebraically that the determinant function is multiplicative in the 2x2 case.<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 #17233 by </B><A HREF=/tutors/aboutme.mpl?userid=mbarugel><B>mbarugel(146)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mbarugel><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=17233\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll solve here the general case. We have two matrices:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2Ca1%2Ca2%2Ca3%2Ca4%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2Ca1%2Ca2%2Ca3%2Ca4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=%28matrix%282%2C2%2Cb1%2Cb2%2Cb3%2Cb4%29%29 ALIGN=MIDDLE  ALT=\"B=%28matrix%282%2C2%2Cb1%2Cb2%2Cb3%2Cb4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then we have that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=det%28A%29=a1a4-a3a2 ALIGN=MIDDLE  ALT=\"det%28A%29=a1a4-a3a2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=det%28B%29=b1b4-b3b2 ALIGN=MIDDLE  ALT=\"det%28B%29=b1b4-b3b2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now consider the multiplication A*B. The result of the multiplication is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2AB=%28matrix%282%2C2%2Ca1b1+%2B+a2b3%2C+a1b2+%2B+a2b4%2C+a3b1%2Ba4b3%2Ca3b2%2Ba4b4%29%29 ALIGN=MIDDLE  ALT=\"A%2AB=%28matrix%282%2C2%2Ca1b1+%2B+a2b3%2C+a1b2+%2B+a2b4%2C+a3b1%2Ba4b3%2Ca3b2%2Ba4b4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's find the determinant of this matrix:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=det%28A%2AB%29=+%28a1b1+%2B+a2b3%29%2A%28a3b2%2Ba4b4%29+-+%28a3b1%2Ba4b3%29%2A%28a1b2+%2B+a2b4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "After applying the distributive property to each multiplication, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a1b1a3b2+%2B+a1b1a4b4+%2B+a2b3a3b2+%2B+a2b3a4b4+-+a3b1a1b2+-+a3b1a2b4+-+a4b3a1b2+-+a4b3a2b4 ALIGN=MIDDLE  ALT=\"a1b1a3b2+%2B+a1b1a4b4+%2B+a2b3a3b2+%2B+a2b3a4b4+-+a3b1a1b2+-+a3b1a2b4+-+a4b3a1b2+-+a4b3a2b4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But some of these terms cancel each other out. Specifically, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a1b1a3b2+-+a3b1a1b2+=+0 ALIGN=MIDDLE  ALT=\"a1b1a3b2+-+a3b1a1b2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a2b3a4b4+-+a4b3a2b4+=+0 ALIGN=MIDDLE  ALT=\"a2b3a4b4+-+a4b3a2b4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We're left with \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=det%28A%2AB%29=a1b1a4b4+%2B+a2b3a3b2+-+a3b1a2b4+-+a4b3a1b2 ALIGN=MIDDLE  ALT=\"det%28A%2AB%29=a1b1a4b4+%2B+a2b3a3b2+-+a3b1a2b4+-+a4b3a1b2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, let's check that this is the same as det(A)*det(B):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=det%28A%29%2Adet%28B%29+=+%28a1a4-a3a2%29%28b1b4-b3b2%29 ALIGN=MIDDLE  ALT=\"det%28A%29%2Adet%28B%29+=+%28a1a4-a3a2%29%28b1b4-b3b2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Applying distributive property:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=det%28A%29%2Adet%28B%29+=+a1a4b1b4+-+a1a4b3b2+-+a3a2b1b4+%2B+a3a2b3b2 ALIGN=MIDDLE  ALT=\"det%28A%29%2Adet%28B%29+=+a1a4b1b4+-+a1a4b3b2+-+a3a2b1b4+%2B+a3a2b3b2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now compare the expressions we found for det(A*B) and for det(A)*det(B). Rearranging some of the factors in each term, they are exactly the same. \r<BR>\n" );
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document.write( "I hope this helps!<BR>\n" );
document.write( "Get more answers at <a  rel=nofollow HREF=http://www.onlinemathanswers.com>Online Math Answers.com</a>! </SPAN>\n" );
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