document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161541>Question 1161541</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the matrix shown below: Find the determinant of the matrix Q.\r<BR>\n" );
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document.write( "A. -67 \r<BR>\n" );
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document.write( "B. -65 \r<BR>\n" );
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document.write( "C. 65 \r<BR>\n" );
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document.write( "D. 67\r<BR>\n" );
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document.write( "The question is provided in the link here: https://i.imgur.com/IKY3rQg.png </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #785155 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=785155\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I certainly wouldn't go to all the work that tutor @MathLover1 did to find the determinant.  There is a lot of work she didn't show involved in turning the entries below the main diagonal to zeros.<br><BR>\n" );
document.write( "For a 3x3 matrix, expanding by minors is one way to find the determinant.<br><BR>\n" );
document.write( "Your matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C3%2C2%2C3%2C4%2C-3%2C2%2C1%2C5%2C-1%2C6%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C3%2C2%2C3%2C4%2C-3%2C2%2C1%2C5%2C-1%2C6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Expanding by minors, the determinant is<br><BR>\n" );
document.write( "2 times the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C2%2C1%2C-1%2C6%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C2%2C1%2C-1%2C6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "minus 3 times the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C-3%2C1%2C5%2C6%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C-3%2C1%2C5%2C6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "plus 4 times the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C2%2C-3%2C2%2C5%2C-1%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C2%2C-3%2C2%2C5%2C-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2812%2B1%29-3%28-18-5%29%2B4%283-10%29+=+26%2B69-28+=+67 ALIGN=MIDDLE  ALT=\"2%2812%2B1%29-3%28-18-5%29%2B4%283-10%29+=+26%2B69-28+=+67\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Answer D<br><BR>\n" );
document.write( "For larger matrices, expanding by minors is very inefficient.  Here is another method for calculating the determinant of any square matrix.<br><BR>\n" );
document.write( "(1) Create a new matrix by copying (for a 3x3 matrix) the first two columns to make a 3x5 matrix:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C5%2C2%2C3%2C4%2C2%2C3%2C-3%2C2%2C1%2C-3%2C2%2C5%2C-1%2C6%2C5%2C-1%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C5%2C2%2C3%2C4%2C2%2C3%2C-3%2C2%2C1%2C-3%2C2%2C5%2C-1%2C6%2C5%2C-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) The determinant of the original matrix is<BR>\n" );
document.write( "the sum of the products of the entries in the down-to-the-right diagonals<BR>\n" );
document.write( "minus<BR>\n" );
document.write( "the sum of the products of the entries on the down-to-the-left diagonals:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282%2A2%2A6%29%2B%283%2A1%2A5%29%2B%284%28-3%29%28-1%29%29%29 ALIGN=MIDDLE  ALT=\"%28%282%2A2%2A6%29%2B%283%2A1%2A5%29%2B%284%28-3%29%28-1%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "minus<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%284%2A2%2A5%29%2B%282%2A1%28-1%29%29%2B%283%28-3%29%2A6%29%29 ALIGN=MIDDLE  ALT=\"%28%284%2A2%2A5%29%2B%282%2A1%28-1%29%29%2B%283%28-3%29%2A6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2824%2B15%2B12%29-%2840-2-54%29+=+51-%28-16%29+=+67 ALIGN=MIDDLE  ALT=\"%2824%2B15%2B12%29-%2840-2-54%29+=+51-%28-16%29+=+67\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
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