document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=380278>Question 380278</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you help me solve the system of equations using matrices: x-2y+3z=7   2x+y+z=5   -3x+2y-2z=-10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #269889 by </B><A HREF=/tutors/aboutme.mpl?userid=Jk22><B>Jk22(389)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Jk22><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=Jk22><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=269889\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=cramers_rule_3x3.solver>Using Cramer's Rule to Solve Systems with 3 variables</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%281%2Ax%2B-2%2Ay%2B3%2Az=7%2C2%2Ax%2B1%2Ay%2B1%2Az=5%2C-3%2Ax%2B2%2Ay%2B-2%2Az=-10%29 ALIGN=MIDDLE  ALT=\"system%281%2Ax%2B-2%2Ay%2B3%2Az=7%2C2%2Ax%2B1%2Ay%2B1%2Az=5%2C-3%2Ax%2B2%2Ay%2B-2%2Az=-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  First let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is the matrix formed by the coefficients of the given system of equations.<BR>\n" );
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document.write( "  Take note that the right hand values of the system are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and they are highlighted here: <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%281%2Ax%2B-2%2Ay%2B3%2Az=highlight%287%29%2C2%2Ax%2B1%2Ay%2B1%2Az=highlight%285%29%2C-3%2Ax%2B2%2Ay%2B-2%2Az=highlight%28-10%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  These values are important as they will be used to replace the columns of the matrix A.<BR>\n" );
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document.write( "  Now let's calculate the the <a  rel=nofollow HREF=http://en.wikipedia.org/wiki/Determinant>determinant</a> of the matrix A to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=15 ALIGN=MIDDLE  ALT=\"abs%28A%29=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>solver</a>.<BR>\n" );
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document.write( "  Notation note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29 ALIGN=MIDDLE  ALT=\"abs%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > denotes the determinant of the matrix A.<BR>\n" );
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document.write( "  Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since we're replacing the 'x' column so to speak).<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%283%2C3%2Chighlight%287%29%2C-2%2C3%2Chighlight%285%29%2C1%2C1%2Chighlight%28-10%29%2C2%2C-2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Bx%5D%29=32 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=32\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Again, as a space saver, I didn't include the calculations of the determinant. Check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>solver</a> to see how to find this determinant.<BR>\n" );
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document.write( "  To find the first solution, simply divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%2832%29%2F%2815%29=32%2F15 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%2832%29%2F%2815%29=32%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the first solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=32%2F15 ALIGN=MIDDLE  ALT=\"x=32%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  We'll follow the same basic idea to find the other two solutions. Let's reset by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > again (this is the coefficient matrix).<BR>\n" );
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document.write( "  Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since we're replacing the 'y' column in a way).<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%283%2C3%2C1%2Chighlight%287%29%2C3%2C2%2Chighlight%285%29%2C1%2C-3%2Chighlight%28-10%29%2C-2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5By%5D%29=-8 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  To find the second solution, divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-8%29%2F%2815%29=-8%2F15 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-8%29%2F%2815%29=-8%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the second solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-8%2F15 ALIGN=MIDDLE  ALT=\"y=-8%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Let's reset again by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the coefficient matrix.<BR>\n" );
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document.write( "  Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D ALIGN=MIDDLE  ALT=\"A%5Bz%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D=%28matrix%283%2C3%2C1%2C-2%2Chighlight%287%29%2C2%2C1%2Chighlight%285%29%2C-3%2C2%2Chighlight%28-10%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D ALIGN=MIDDLE  ALT=\"A%5Bz%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Bz%5D%29=19 ALIGN=MIDDLE  ALT=\"abs%28A%5Bz%5D%29=19\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  To find the third solution, divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bz%5D ALIGN=MIDDLE  ALT=\"A%5Bz%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%2819%29%2F%2815%29=19%2F15 ALIGN=MIDDLE  ALT=\"z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%2819%29%2F%2815%29=19%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the third solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=19%2F15 ALIGN=MIDDLE  ALT=\"z=19%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Final Answer:<BR>\n" );
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document.write( "  So the three solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=32%2F15 ALIGN=MIDDLE  ALT=\"x=32%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-8%2F15 ALIGN=MIDDLE  ALT=\"y=-8%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=19%2F15 ALIGN=MIDDLE  ALT=\"z=19%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the ordered triple (32/15, -8/15, 19/15)<BR>\n" );
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document.write( "  Note: there is a lot of work that is hidden in finding the determinants. Take a look at this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-3x3-matrix.solver>3x3 Determinant Solver</a> to see how to get each determinant.<BR>\n" );
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document.write( "det(A) is <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=determinant_of_3x3_matrix.solver>Finding the Determinant of a 3x3 Matrix</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><br>If you have the general 3x3 matrix:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>the <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Determinant#3-by-3_matrices\">determinant</a> is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%2Aabs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29-b%2Aabs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29%2Bc%2Aabs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Which further breaks down to:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are determinants themselves.<br> If you need help finding the determinant of 2x2 matrices (which is <i>required</i> to find the determinant of 3x3 matrices), check out this <a  rel=nofollow HREF=\"http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver\">solver</a><br><br>--------------------------------------------------------------<br><br><br>From the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3 ALIGN=MIDDLE  ALT=\"c=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=2 ALIGN=MIDDLE  ALT=\"d=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=1 ALIGN=MIDDLE  ALT=\"e=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=1 ALIGN=MIDDLE  ALT=\"f=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=-3 ALIGN=MIDDLE  ALT=\"g=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=-2 ALIGN=MIDDLE  ALT=\"i=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general 3x3 determinant.<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=%281%29%28%281%29%28-2%29-%281%29%282%29%29-%28-2%29%28%282%29%28-2%29-%281%29%28-3%29%29%2B%283%29%28%282%29%282%29-%281%29%28-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given values (see above)<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=%281%29%28-2-2%29-%28-2%29%28-4--3%29%2B%283%29%284--3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=%281%29%28-4%29-%28-2%29%28-1%29%2B%283%29%287%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=-4-2%2B21 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=-4-2%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=15 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.<br><br><br>======================================================================<br><br><br>Answer:<br><br>So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=15 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  which means that the determinant of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C-2%2C3%2C2%2C1%2C1%2C-3%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font color=\"red\">15</font><br></TD></TR>\n" );
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document.write( "det(Ax) is <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=determinant_of_3x3_matrix.solver>Finding the Determinant of a 3x3 Matrix</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><br>If you have the general 3x3 matrix:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>the <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Determinant#3-by-3_matrices\">determinant</a> is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%2Aabs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29-b%2Aabs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29%2Bc%2Aabs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Which further breaks down to:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are determinants themselves.<br> If you need help finding the determinant of 2x2 matrices (which is <i>required</i> to find the determinant of 3x3 matrices), check out this <a  rel=nofollow HREF=\"http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver\">solver</a><br><br>--------------------------------------------------------------<br><br><br>From the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=7 ALIGN=MIDDLE  ALT=\"a=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3 ALIGN=MIDDLE  ALT=\"c=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=5 ALIGN=MIDDLE  ALT=\"d=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=1 ALIGN=MIDDLE  ALT=\"e=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=1 ALIGN=MIDDLE  ALT=\"f=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=-10 ALIGN=MIDDLE  ALT=\"g=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=-2 ALIGN=MIDDLE  ALT=\"i=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general 3x3 determinant.<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=%287%29%28%281%29%28-2%29-%281%29%282%29%29-%28-2%29%28%285%29%28-2%29-%281%29%28-10%29%29%2B%283%29%28%285%29%282%29-%281%29%28-10%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given values (see above)<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=%287%29%28-2-2%29-%28-2%29%28-10--10%29%2B%283%29%2810--10%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=%287%29%28-4%29-%28-2%29%280%29%2B%283%29%2820%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=-28-0%2B60 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=-28-0%2B60\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=32 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.<br><br><br>======================================================================<br><br><br>Answer:<br><br>So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=32 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29=32\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  which means that the determinant of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C7%2C-2%2C3%2C5%2C1%2C1%2C-10%2C2%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font color=\"red\">32</font><br></TD></TR>\n" );
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document.write( "det(Ay) is <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=determinant_of_3x3_matrix.solver>Finding the Determinant of a 3x3 Matrix</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><br>If you have the general 3x3 matrix:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>the <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Determinant#3-by-3_matrices\">determinant</a> is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%2Aabs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29-b%2Aabs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29%2Bc%2Aabs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Which further breaks down to:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are determinants themselves.<br> If you need help finding the determinant of 2x2 matrices (which is <i>required</i> to find the determinant of 3x3 matrices), check out this <a  rel=nofollow HREF=\"http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver\">solver</a><br><br>--------------------------------------------------------------<br><br><br>From the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7 ALIGN=MIDDLE  ALT=\"b=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3 ALIGN=MIDDLE  ALT=\"c=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=2 ALIGN=MIDDLE  ALT=\"d=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=5 ALIGN=MIDDLE  ALT=\"e=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=1 ALIGN=MIDDLE  ALT=\"f=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=-3 ALIGN=MIDDLE  ALT=\"g=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-10 ALIGN=MIDDLE  ALT=\"h=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=-2 ALIGN=MIDDLE  ALT=\"i=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general 3x3 determinant.<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=%281%29%28%285%29%28-2%29-%281%29%28-10%29%29-%287%29%28%282%29%28-2%29-%281%29%28-3%29%29%2B%283%29%28%282%29%28-10%29-%285%29%28-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given values (see above)<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=%281%29%28-10--10%29-%287%29%28-4--3%29%2B%283%29%28-20--15%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=%281%29%280%29-%287%29%28-1%29%2B%283%29%28-5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=0--7%2B-15 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=0--7%2B-15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=-8 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.<br><br><br>======================================================================<br><br><br>Answer:<br><br>So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=-8 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  which means that the determinant of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C7%2C3%2C2%2C5%2C1%2C-3%2C-10%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font color=\"red\">-8</font><br></TD></TR>\n" );
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document.write( "det(Az) is <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=determinant_of_3x3_matrix.solver>Finding the Determinant of a 3x3 Matrix</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><br>If you have the general 3x3 matrix:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>the <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Determinant#3-by-3_matrices\">determinant</a> is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%2Aabs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29-b%2Aabs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29%2Bc%2Aabs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Which further breaks down to:<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are determinants themselves.<br> If you need help finding the determinant of 2x2 matrices (which is <i>required</i> to find the determinant of 3x3 matrices), check out this <a  rel=nofollow HREF=\"http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver\">solver</a><br><br>--------------------------------------------------------------<br><br><br>From the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=7 ALIGN=MIDDLE  ALT=\"c=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=2 ALIGN=MIDDLE  ALT=\"d=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=1 ALIGN=MIDDLE  ALT=\"e=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=5 ALIGN=MIDDLE  ALT=\"f=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=-3 ALIGN=MIDDLE  ALT=\"g=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=-10 ALIGN=MIDDLE  ALT=\"i=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general 3x3 determinant.<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=%281%29%28%281%29%28-10%29-%285%29%282%29%29-%28-2%29%28%282%29%28-10%29-%285%29%28-3%29%29%2B%287%29%28%282%29%282%29-%281%29%28-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given values (see above)<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=%281%29%28-10-10%29-%28-2%29%28-20--15%29%2B%287%29%284--3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=%281%29%28-20%29-%28-2%29%28-5%29%2B%287%29%287%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=-20-10%2B49 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=-20-10%2B49\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=19 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.<br><br><br>======================================================================<br><br><br>Answer:<br><br>So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=19 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29=19\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  which means that the determinant of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C-2%2C7%2C2%2C1%2C5%2C-3%2C2%2C-10%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font color=\"red\">19</font><br></TD></TR>\n" );
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