document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=200660>Question 200660</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi all I asked a similiar question recently and got a reaaly good response, but this imiliar problem is still giving me trouble.<BR>\n" );
document.write( "I need to solve the following Matrices and comment on the outcomes.For the following:<BR>\n" );
document.write( "A = <BR>\n" );
document.write( "(2 3 0)<BR>\n" );
document.write( "1 4 -3 <BR>\n" );
document.write( "B = <BR>\n" );
document.write( "(1 -1 4) <BR>\n" );
document.write( " 2  3 5 <BR>\n" );
document.write( "C = <BR>\n" );
document.write( "(2 -1)<BR>\n" );
document.write( " 3 5 <BR>\n" );
document.write( "Find and state why or why not the following is possible.<BR>\n" );
document.write( "i) A+B<BR>\n" );
document.write( "ii) C^-1<BR>\n" );
document.write( "iii) AB<BR>\n" );
document.write( "1v) B^C </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #151046 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=151046\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <a name=\"top\">\r<BR>\n" );
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document.write( "Table of Contents:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"#1\">Part i)</a><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#2\">Part ii)</a> <BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#3\">Part iii)</a><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"#4\">Part iv)</a>\r<BR>\n" );
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document.write( "Given Matrices:\r<BR>\n" );
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document.write( "Part i)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB=%28matrix%282%2C3%2C2%2C3%2C0%2C1%2C4%2C-3%29%29%2B%28matrix%282%2C3%2C1%2C-1%2C4%2C2%2C3%2C5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the addition of the two given matrices\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB=%28matrix%282%2C3%2C2%2B1%2C3%2B%28-1%29%2C0%2B4%2C1%2B2%2C4%2B3%2C-3%2B5%29%29 ALIGN=MIDDLE  ALT=\"A%2BB=%28matrix%282%2C3%2C2%2B1%2C3%2B%28-1%29%2C0%2B4%2C1%2B2%2C4%2B3%2C-3%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add up the matrices by adding the corresponding entries\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB=%28matrix%282%2C3%2C3%2C2%2C4%2C3%2C7%2C2%29%29 ALIGN=MIDDLE  ALT=\"A%2BB=%28matrix%282%2C3%2C3%2C2%2C4%2C3%2C7%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\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%2BB=%28matrix%282%2C3%2C3%2C2%2C4%2C3%2C7%2C2%29%29 ALIGN=MIDDLE  ALT=\"A%2BB=%28matrix%282%2C3%2C3%2C2%2C4%2C3%2C7%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Part ii)\r<BR>\n" );
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document.write( "To find the inverse of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29 ALIGN=MIDDLE  ALT=\"C=%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can follow these steps:\r<BR>\n" );
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document.write( "Step 1) Find the determinant\r<BR>\n" );
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document.write( "The <a  rel=nofollow HREF=\"http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver\">determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29=13 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=13 ALIGN=MIDDLE  ALT=\"d=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "note: since the determinant is NOT equal to zero, the matrix inverse exists.\r<BR>\n" );
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document.write( "Step 2) Swap the values\r<BR>\n" );
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document.write( "Now switch the highlighted values <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Chighlight%282%29%2C-1%2C3%2Chighlight%285%29%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Chighlight%282%29%2C-1%2C3%2Chighlight%285%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Chighlight%285%29%2C-1%2C3%2Chighlight%282%29%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Chighlight%285%29%2C-1%2C3%2Chighlight%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 3) Change the sign\r<BR>\n" );
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document.write( "Now change the sign of the highlighted values <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2Chighlight%28-1%29%2Chighlight%283%29%2C2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2Chighlight%28-1%29%2Chighlight%283%29%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2Chighlight%281%29%2Chighlight%28-3%29%2C2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2Chighlight%281%29%2Chighlight%28-3%29%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 4) Multiply by the inverse of the determinant\r<BR>\n" );
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document.write( "Multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fd ALIGN=MIDDLE  ALT=\"1%2Fd\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fd%29%28matrix%282%2C2%2C5%2C1%2C-3%2C2%29%29 ALIGN=MIDDLE  ALT=\"%281%2Fd%29%28matrix%282%2C2%2C5%2C1%2C-3%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=13 ALIGN=MIDDLE  ALT=\"d=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F13%29%28matrix%282%2C2%2C5%2C1%2C-3%2C2%29%29 ALIGN=MIDDLE  ALT=\"%281%2F13%29%28matrix%282%2C2%2C5%2C1%2C-3%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 5) Perform scalar multiplication and simplify.\r<BR>\n" );
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document.write( "Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F13 ALIGN=MIDDLE  ALT=\"1%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" > by EVERY element to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C%281%2F13%29%285%29%2C%281%2F13%29%281%29%2C%281%2F13%29%28-3%29%2C%281%2F13%29%282%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2F13%2C1%2F13%2C-3%2F13%2C2%2F13%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2F13%2C1%2F13%2C-3%2F13%2C2%2F13%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 the inverse of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2F13%2C1%2F13%2C-3%2F13%2C2%2F13%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2F13%2C1%2F13%2C-3%2F13%2C2%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This means that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29 ALIGN=MIDDLE  ALT=\"C=%28matrix%282%2C2%2C2%2C-1%2C3%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5E%28-1%29=%28matrix%282%2C2%2C5%2F13%2C1%2F13%2C-3%2F13%2C2%2F13%29%29 ALIGN=MIDDLE  ALT=\"C%5E%28-1%29=%28matrix%282%2C2%2C5%2F13%2C1%2F13%2C-3%2F13%2C2%2F13%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "note: to verify your work, multiply the matrices <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5E%28-1%29 ALIGN=MIDDLE  ALT=\"C%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you should get the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the identity matrix).\r<BR>\n" );
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document.write( "Part iii)\r<BR>\n" );
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document.write( "Take note that the dimensions of matrices A and B are 2x3 and 2x3 respectively. Since the inner dimensions do NOT match, this means that the matrix product AB is NOT defined. In other words, AB does NOT exist.\r<BR>\n" );
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document.write( "note: if A and B are mxn and nxp matrices, then the product AB is defined. It is only defined if the number of columns of A equals the number of rows of B. Otherwise, AB does NOT exist.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the matrix product AB is not defined (ie it doesn't exist).\r<BR>\n" );
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document.write( "Part iv)\r<BR>\n" );
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document.write( "I'm sure you meant to write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC ALIGN=MIDDLE  ALT=\"B%5ET%2AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > correct? If so, then...\r<BR>\n" );
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document.write( "Step 1: Transpose matrix B to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET ALIGN=MIDDLE  ALT=\"B%5ET\"   DISABLED_style= \"max-width:90%; height:auto;\" >. You do this by converting each row into a column (and vice versa).\r<BR>\n" );
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document.write( "Step 2: Multiply matrices <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET ALIGN=MIDDLE  ALT=\"B%5ET\"   DISABLED_style= \"max-width:90%; height:auto;\" > and C. Since the inner dimensions are equal, this means that the matrix product <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC ALIGN=MIDDLE  ALT=\"B%5ET%2AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is defined.\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( "note: the \"x\"s are just placeholders for now\r<BR>\n" );
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document.write( "--------------------------------------------------\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">1st</font> row of the first matrix by the <font size=\"4\" color=\"green\">1st</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font size=\"4\" color=\"red\">1st</font> row, <font size=\"4\" color=\"green\">1st</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2Chighlight%281%29%2Chighlight%282%29%2C-1%2C3%2C4%2C5%29%29%28matrix%282%2C2%2Chighlight%282%29%2C-1%2Chighlight%283%29%2C5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%282%29%2B%282%29%2A%283%29=2%2B6=8 ALIGN=MIDDLE  ALT=\"%281%29%2A%282%29%2B%282%29%2A%283%29=2%2B6=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( " So the element in the <font size=\"4\" color=\"red\">1st</font> row, <font size=\"4\" color=\"green\">1st</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C8%2Cx%2Cx%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"B%5ET%2AC=%28matrix%283%2C2%2C8%2Cx%2Cx%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">1st</font> row of the first matrix by the <font size=\"4\" color=\"green\">2nd</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "\n" );
document.write( "<font size=\"4\" color=\"red\">1st</font> row, <font size=\"4\" color=\"green\">2nd</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2Chighlight%281%29%2Chighlight%282%29%2C-1%2C3%2C4%2C5%29%29%28matrix%282%2C2%2C2%2Chighlight%28-1%29%2C3%2Chighlight%285%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%28-1%29%2B%282%29%2A%285%29=-1%2B10=9 ALIGN=MIDDLE  ALT=\"%281%29%2A%28-1%29%2B%282%29%2A%285%29=-1%2B10=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( " So the element in the <font size=\"4\" color=\"red\">1st</font> row, <font size=\"4\" color=\"green\">2nd</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2Cx%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2Cx%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( "================================================================================\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">2nd</font> row of the first matrix by the <font size=\"4\" color=\"green\">1st</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "<font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">1st</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C1%2C2%2Chighlight%28-1%29%2Chighlight%283%29%2C4%2C5%29%29%28matrix%282%2C2%2Chighlight%282%29%2C-1%2Chighlight%283%29%2C5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2A%282%29%2B%283%29%2A%283%29=-2%2B9=7 ALIGN=MIDDLE  ALT=\"%28-1%29%2A%282%29%2B%283%29%2A%283%29=-2%2B9=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( " So the element in the <font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">1st</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2C7%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2C7%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">2nd</font> row of the first matrix by the <font size=\"4\" color=\"green\">2nd</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "\n" );
document.write( "<font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">2nd</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C1%2C2%2Chighlight%28-1%29%2Chighlight%283%29%2C4%2C5%29%29%28matrix%282%2C2%2C2%2Chighlight%28-1%29%2C3%2Chighlight%285%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2A%28-1%29%2B%283%29%2A%285%29=1%2B15=16 ALIGN=MIDDLE  ALT=\"%28-1%29%2A%28-1%29%2B%283%29%2A%285%29=1%2B15=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( " So the element in the <font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">2nd</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2C7%2C16%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2C7%2C16%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "================================================================================\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">3rd</font> row of the first matrix by the <font size=\"4\" color=\"green\">1st</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "\n" );
document.write( "<font size=\"4\" color=\"red\">3rd</font> row, <font size=\"4\" color=\"green\">1st</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C1%2C2%2C-1%2C3%2Chighlight%284%29%2Chighlight%285%29%29%29%28matrix%282%2C2%2Chighlight%282%29%2C-1%2Chighlight%283%29%2C5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%2A%282%29%2B%285%29%2A%283%29=8%2B15=23 ALIGN=MIDDLE  ALT=\"%284%29%2A%282%29%2B%285%29%2A%283%29=8%2B15=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( " So the element in the <font size=\"4\" color=\"red\">3rd</font> row, <font size=\"4\" color=\"green\">1st</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23 ALIGN=MIDDLE  ALT=\"23\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2C7%2C16%2C23%2Cx%29%29 ALIGN=MIDDLE  ALT=\"B%5ET%2AC=%28matrix%283%2C2%2C8%2C9%2C7%2C16%2C23%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">3rd</font> row of the first matrix by the <font size=\"4\" color=\"green\">2nd</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font size=\"4\" color=\"red\">3rd</font> row, <font size=\"4\" color=\"green\">2nd</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC=%28matrix%283%2C2%2C1%2C2%2C-1%2C3%2Chighlight%284%29%2Chighlight%285%29%29%29%28matrix%282%2C2%2C2%2Chighlight%28-1%29%2C3%2Chighlight%285%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%2A%28-1%29%2B%285%29%2A%285%29=-4%2B25=21 ALIGN=MIDDLE  ALT=\"%284%29%2A%28-1%29%2B%285%29%2A%285%29=-4%2B25=21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( " So the element in the <font size=\"4\" color=\"red\">3rd</font> row, <font size=\"4\" color=\"green\">2nd</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21 ALIGN=MIDDLE  ALT=\"21\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5ET%2AC+=+%28matrix%283%2C2%2C8%2C9%2C7%2C16%2C23%2C21%29%29 ALIGN=MIDDLE  ALT=\"B%5ET%2AC+=+%28matrix%283%2C2%2C8%2C9%2C7%2C16%2C23%2C21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==============================================================================\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=B%5ET%2AC+=+%28matrix%283%2C2%2C1%2C2%2C-1%2C3%2C4%2C5%29%29%2A%28matrix%283%2C2%2C2%2C-1%2C3%2C5%29%29+=+%28matrix%283%2C2%2C8%2C9%2C7%2C16%2C23%2C21%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words,\r<BR>\n" );
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