document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=200482>Question 200482</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi everyone I asked a similiar question recently but got no responce, I may have input the question incorrectly.<BR>\n" );
document.write( "I  need to solve the following Matrices and comment on the outcomes.<BR>\n" );
document.write( "For the following:<BR>\n" );
document.write( "A =  <BR>\n" );
document.write( "( 2 -1  1 )<BR>\n" );
document.write( "  1  1  -2\r<BR>\n" );
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document.write( "B =  <BR>\n" );
document.write( "( 1 -1 ) <BR>\n" );
document.write( "  2  3 \r<BR>\n" );
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document.write( "C =  <BR>\n" );
document.write( "( 1 -1 )<BR>\n" );
document.write( "  0  1<BR>\n" );
document.write( " -2  1<BR>\n" );
document.write( " <BR>\n" );
document.write( "Find and state why or why not the following is possible.<BR>\n" );
document.write( "i) A^T + C<BR>\n" );
document.write( "ii) AB<BR>\n" );
document.write( "iii) CB </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150764 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=150764\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C3%2C2%2C-1%2C1%2C1%2C1%2C-2%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C3%2C2%2C-1%2C1%2C1%2C1%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=%28matrix%283%2C2%2C1%2C-1%2C0%2C1%2C-2%2C1%29%29 ALIGN=MIDDLE  ALT=\"C=%28matrix%283%2C2%2C1%2C-1%2C0%2C1%2C-2%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "i)\r<BR>\n" );
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document.write( "Step 1: Find the transpose of A to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5ET ALIGN=MIDDLE  ALT=\"A%5ET\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5ET=%28matrix%282%2C3%2C2%2C-1%2C1%2C1%2C1%2C-2%29%29%5ET=%28matrix%283%2C2%2C2%2C1%2C-1%2C1%2C1%2C-2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 2: Add  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5ET ALIGN=MIDDLE  ALT=\"A%5ET\"   DISABLED_style= \"max-width:90%; height:auto;\" > to C (just add up each component) to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5ET%2BC=%28matrix%283%2C2%2C2%2C1%2C-1%2C1%2C1%2C-2%29%29%2B%28matrix%283%2C2%2C1%2C-1%2C0%2C1%2C-2%2C1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5ET%2BC=%28matrix%283%2C2%2C3%2C0%2C-1%2C2%2C-1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A%5ET%2BC=%28matrix%283%2C2%2C3%2C0%2C-1%2C2%2C-1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5ET%2BC=%28matrix%283%2C2%2C3%2C0%2C-1%2C2%2C-1%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A%5ET%2BC=%28matrix%283%2C2%2C3%2C0%2C-1%2C2%2C-1%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "ii) The product AB is NOT possible since the dimensions of A are 2x3 and the dimensions of B are 2x2. The inner dimensions MUST match for matrix multiplication to be defined.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not exist.\r<BR>\n" );
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document.write( "iii)\r<BR>\n" );
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document.write( "Since the first matrix is a 3 by 2 matrix and the second matrix is a 2 by 2 matrix, this means that the resulting matrix will be a 3 by  2 matrix.\r<BR>\n" );
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document.write( "So the final resulting matrix will look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CB=%28matrix%283%2C2%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"CB=%28matrix%283%2C2%2Cx%2Cx%2Cx%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "note: the \"x\"s are just placeholders for now\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the 1st row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "1st row, 1st column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C2%2Chighlight%281%29%2Chighlight%28-1%29%2C0%2C1%2C-2%2C1%29%29%28matrix%282%2C2%2Chighlight%281%29%2C-1%2Chighlight%282%29%2C3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%281%29%2B%28-1%29%2A%282%29=1%2B-2=-1 ALIGN=MIDDLE  ALT=\"%281%29%2A%281%29%2B%28-1%29%2A%282%29=1%2B-2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the 1st row, 1st column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CB=%28matrix%283%2C2%2C-1%2Cx%2Cx%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"CB=%28matrix%283%2C2%2C-1%2Cx%2Cx%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the 1st row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "1st row, 2nd column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C2%2Chighlight%281%29%2Chighlight%28-1%29%2C0%2C1%2C-2%2C1%29%29%28matrix%282%2C2%2C1%2Chighlight%28-1%29%2C2%2Chighlight%283%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%28-1%29%2B%28-1%29%2A%283%29=-1%2B-3=-4 ALIGN=MIDDLE  ALT=\"%281%29%2A%28-1%29%2B%28-1%29%2A%283%29=-1%2B-3=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the 1st row, 2nd column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CB=%28matrix%283%2C2%2C-1%2C-4%2Cx%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"CB=%28matrix%283%2C2%2C-1%2C-4%2Cx%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the 2nd row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "2nd row, 1st column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C2%2C1%2C-1%2Chighlight%280%29%2Chighlight%281%29%2C-2%2C1%29%29%28matrix%282%2C2%2Chighlight%281%29%2C-1%2Chighlight%282%29%2C3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%2A%281%29%2B%281%29%2A%282%29=0%2B2=2 ALIGN=MIDDLE  ALT=\"%280%29%2A%281%29%2B%281%29%2A%282%29=0%2B2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the 2nd row, 1st column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CB=%28matrix%283%2C2%2C-1%2C-4%2C2%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"CB=%28matrix%283%2C2%2C-1%2C-4%2C2%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the 2nd row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "2nd row, 2nd column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C2%2C1%2C-1%2Chighlight%280%29%2Chighlight%281%29%2C-2%2C1%29%29%28matrix%282%2C2%2C1%2Chighlight%28-1%29%2C2%2Chighlight%283%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%2A%28-1%29%2B%281%29%2A%283%29=0%2B3=3 ALIGN=MIDDLE  ALT=\"%280%29%2A%28-1%29%2B%281%29%2A%283%29=0%2B3=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the 2nd row, 2nd column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the 3rd row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "3rd row, 1st column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C2%2C1%2C-1%2C0%2C1%2Chighlight%28-2%29%2Chighlight%281%29%29%29%28matrix%282%2C2%2Chighlight%281%29%2C-1%2Chighlight%282%29%2C3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%2A%281%29%2B%281%29%2A%282%29=-2%2B2=0 ALIGN=MIDDLE  ALT=\"%28-2%29%2A%281%29%2B%281%29%2A%282%29=-2%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the 3rd row, 1st column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CB=%28matrix%283%2C2%2C-1%2C-4%2C2%2C3%2C0%2Cx%29%29 ALIGN=MIDDLE  ALT=\"CB=%28matrix%283%2C2%2C-1%2C-4%2C2%2C3%2C0%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the 3rd row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "3rd row, 2nd column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C2%2C1%2C-1%2C0%2C1%2Chighlight%28-2%29%2Chighlight%281%29%29%29%28matrix%282%2C2%2C1%2Chighlight%28-1%29%2C2%2Chighlight%283%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%2A%28-1%29%2B%281%29%2A%283%29=2%2B3=5 ALIGN=MIDDLE  ALT=\"%28-2%29%2A%28-1%29%2B%281%29%2A%283%29=2%2B3=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the 3rd row, 2nd column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C2%2C1%2C-1%2C0%2C1%2C-2%2C1%29%29%2A%28matrix%283%2C2%2C1%2C-1%2C2%2C3%29%29=%28matrix%283%2C2%2C-1%2C-4%2C2%2C3%2C0%2C5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CB=%28matrix%283%2C2%2C-1%2C-4%2C2%2C3%2C0%2C5%29%29 ALIGN=MIDDLE  ALT=\"CB=%28matrix%283%2C2%2C-1%2C-4%2C2%2C3%2C0%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );