document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=102575>Question 102575</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help with the following:\r<BR>\n" );
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document.write( "Let A=  2  -1 <BR>\n" );
document.write( "        4   3 <BR>\n" );
document.write( "    B= [3   1]\r<BR>\n" );
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document.write( "Perform the following operations if possible:<BR>\n" );
document.write( "a. 4BA<BR>\n" );
document.write( "b. B^2<BR>\n" );
document.write( "c. A+2B\r<BR>\n" );
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document.write( "Am I right in the possiblity that a and c are not possible? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #74636 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=74636\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=%28matrix%281%2C2%2C3%2C1%29%29 ALIGN=MIDDLE  ALT=\"B=%28matrix%281%2C2%2C3%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"#a\">Part a</a><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#b\">Part b</a><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#c\">Part c</a>\r<BR>\n" );
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document.write( "<a name=\"a\"></a>\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4BA ALIGN=MIDDLE  ALT=\"4BA\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given product\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28matrix%281%2C2%2C3%2C1%29%29%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"4%28matrix%281%2C2%2C3%2C1%29%29%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in matrices A and B\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2C4%2A3%2C4%2A1%29%29%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2C4%2A3%2C4%2A1%29%29%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the scalar number 4 by every entry in matrix A\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2C12%2C4%29%29%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2C12%2C4%29%29%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Now multiply matrices B and A. In order for this to be defined, the number of columns in B needs to be the same as the number of rows in A. Since this is true, BA is defined.\r<BR>\n" );
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document.write( "Now if you have two general matrices \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2Ca%2Cb%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2Ca%2Cb%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Cc%2Cd%2Ce%2Cf%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Cc%2Cd%2Ce%2Cf%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can multiply them like this \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2Ca%2Ac%2Bb%2Ae%2Ca%2Ad%2Bb%2Af%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2Ca%2Ac%2Bb%2Ae%2Ca%2Ad%2Bb%2Af%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which will result in a 1x2 matrix\r<BR>\n" );
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document.write( "So for the matrices \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2C12%2C4%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2C12%2C4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "They can be multiplied like this: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2C12%2A2%2B4%2A4%2C12%2A-1%2B4%2A3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2C12%2A2%2B4%2A4%2C12%2A-1%2B4%2A3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2C24%2B16%2C-12%2B12%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2C24%2B16%2C-12%2B12%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "which adds to \r<BR>\n" );
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document.write( "<a name=\"b\"></a>\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2 ALIGN=MIDDLE  ALT=\"B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2AB ALIGN=MIDDLE  ALT=\"B%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%281%2C2%2C3%2C1%29%29%28matrix%281%2C2%2C3%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%281%2C2%2C3%2C1%29%29%28matrix%281%2C2%2C3%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand\r<BR>\n" );
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document.write( "Since the columns of B do not equal the rows of B, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2 ALIGN=MIDDLE  ALT=\"B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not defined. So in order to perform powers of matrices, the matrix needs to be square\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2B2B ALIGN=MIDDLE  ALT=\"A%2B2B\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29%2B2%28matrix%281%2C2%2C3%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C2%2C-1%2C4%2C3%29%29%2B2%28matrix%281%2C2%2C3%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given matrices\r<BR>\n" );
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document.write( "Since B does not have a second row, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2B2B ALIGN=MIDDLE  ALT=\"A%2B2B\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not defined (remember, when you add two matrices, they need to be the same size) </SPAN>\n" );
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