document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=58434>Question 58434</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please give me a step by step guide on how to solve 2x2 and 3x3 matrices using cramers rules! PLEASE PLEASE PLEASE!!!!! THANKS! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #40022 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=40022\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Write a matrix equation<BR>\n" );
document.write( "equivalent to the system of equations.<BR>\n" );
document.write( "9x + 9y = -9<BR>\n" );
document.write( "5x - 2y = 6<BR>\n" );
document.write( "Cramer's rule.<BR>\n" );
document.write( "6x + 4y = -4<BR>\n" );
document.write( "y = -3x - 7<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 12176 by venugopalramana(1088) About Me on<BR>\n" );
document.write( "2006-01-03 10:24:38 (Show Source):<BR>\n" );
document.write( "Write a matrix equation equivalent to the system of<BR>\n" );
document.write( "equations.<BR>\n" );
document.write( "9x + 9y = -9<BR>\n" );
document.write( "5x - 2y = 6<BR>\n" );
document.write( "we write the matrix equation as (A)*(X)=(C)..where A<BR>\n" );
document.write( "is the (2,2)matrix of coefficients namely,9,9,5 and -2<BR>\n" );
document.write( "here.X is the (2,1) matrix of unknowns x and y and C<BR>\n" );
document.write( "is the constants (2,1)matrix on the right side of the<BR>\n" );
document.write( "eqn NAMELY -9 AND 6.you will find that from the rule<BR>\n" );
document.write( "for equality of matrices,the above matrix eqn.in<BR>\n" );
document.write( "effect means the same as that of the given equations.<BR>\n" );
document.write( "(matrix(2,2,9,9,5,-2))*(matrix(2,1,x,y))=(matrix(2,1,-9,6))<BR>\n" );
document.write( "Cramer's rule.<BR>\n" );
document.write( "6x + 4y = -4<BR>\n" );
document.write( "y = -3x - 7<BR>\n" );
document.write( "GIVING BELOW EXAMPE OF CRAMERS RULE.<BR>\n" );
document.write( "My question is.... I was wondering how to do the<BR>\n" );
document.write( "cramer rule on a 3x3. I have found a bunch of examples<BR>\n" );
document.write( "and stuff, but I want to know how in the world do you<BR>\n" );
document.write( "find the determinants of the D, Dx, Dy.and Dz. If you<BR>\n" );
document.write( "could just tell me how, that would be great.<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 9496 by venugopalramana(585) About Me on<BR>\n" );
document.write( "2005-11-15 10:49:54 (Show Source):<BR>\n" );
document.write( "SEE THE FOLLOWING AND COME BACK IF YOU HAVE<BR>\n" );
document.write( "DIFFICULTY.HERE C,CX,CY,CZ REFER TO YOUR<BR>\n" );
document.write( "D,DX,DY,DZ...JUST A DIFFERENCE IN NOMENCLATURE.I<BR>\n" );
document.write( "SHOWED IN DETAIL A 2X2 DETERMINANT AND THEN IN BRIEF A<BR>\n" );
document.write( "3X3 DETERMINANT<BR>\n" );
document.write( "2x+y=4<BR>\n" );
document.write( "3x-y=6<BR>\n" );
document.write( "make a deteminant with coefficients of x (2,3)and<BR>\n" );
document.write( "y(1,-1) in the 2 eqns.call it C.(Actually for a<BR>\n" );
document.write( "determinant as you know ,the numbers are contained in<BR>\n" );
document.write( "vertical bars at either end like |xx|,but in the<BR>\n" );
document.write( "following the bars are omitted due to difficulty in<BR>\n" );
document.write( "depiction.you may assume the bars are present)<BR>\n" );
document.write( "C=matrix(2,2,2,1,3,-1)=2*(-1)-(1*3)=-5<BR>\n" );
document.write( "..now use the constants (4,6)to replace coefficients<BR>\n" );
document.write( "of x(2,3) in the above determinant C...call it CX..<BR>\n" );
document.write( "CX=matrix(2,2,4,1,6,-1)=4*(-1)-1*6=-4-6=-10<BR>\n" );
document.write( "..now use the constants (4,6)to replace coefficients<BR>\n" );
document.write( "of y(1,-1) in the above determinant C...call it CY..<BR>\n" );
document.write( "CY=matrix(2,2,2,4,3,6)=2*6-3*4=12=12=0<BR>\n" );
document.write( "..now cramers rule says that<BR>\n" );
document.write( "(x/CX)=(y/CY)=(1/C)..so we get<BR>\n" );
document.write( "x/(-10)=y/0=1/-5<BR>\n" );
document.write( "x=-10/-5=10/5=2<BR>\n" );
document.write( "y=0/-5=0<BR>\n" );
document.write( "************************************<BR>\n" );
document.write( "so using the above method you can do the next problem<BR>\n" );
document.write( "..here due to presence of 3 variables you will get<BR>\n" );
document.write( "3rd.order determinants...4 in all...namely C,CX,CY and<BR>\n" );
document.write( "CZ,the last formula also extends to include z ,<BR>\n" );
document.write( "(x/CX)=(y/CY)=(z/CZ)=(1/C)..<BR>\n" );
document.write( "but the procedure is same ..<BR>\n" );
document.write( "2x+3y+ z= 5<BR>\n" );
document.write( "x+y-2z= -2<BR>\n" );
document.write( "-3x +z=-7 ...<BR>\n" );
document.write( "...just to give you the idea<BR>\n" );
document.write( "C=matrix(3,3,2,3,1,1,1,-2,-3,0,1)..and<BR>\n" );
document.write( "CZ=matrix(3,3,2,3,5,1,1,-2,-3,0,7)..etc..hope you can<BR>\n" );
document.write( "work out the rest\r<BR>\n" );
document.write( "\n" );
document.write( "Matrices-and-determiminant/33609: solve each system\r<BR>\n" );
document.write( "\n" );
document.write( "x+y-z= -1<BR>\n" );
document.write( "4x-3y+2z= 16<BR>\n" );
document.write( "2x-2y-3z=5<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 20008 by venugopalramana(1619) About Me  on<BR>\n" );
document.write( "2006-04-14 00:40:08 (Show Source):<BR>\n" );
document.write( "Matrices-and-determiminant/21154: Can you help me to<BR>\n" );
document.write( "solve this Matrix Transformation problem please?<BR>\n" );
document.write( "x-2y+3z=3<BR>\n" );
document.write( "2x+y+5z=8<BR>\n" );
document.write( "3x-y-3z=-22<BR>\n" );
document.write( "Thank You<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 12177 by venugopalramana(1088) About Me on<BR>\n" );
document.write( "2006-01-03 10:32:42 (Show Source):<BR>\n" );
document.write( "SEE THE FOLLOWING EXAMPLE ND TRY.IN CASE OF DIFFICULTY<BR>\n" );
document.write( "PLEASE COME BACK<BR>\n" );
document.write( "Hi, I'm in homeschooling and I'm having trouble with<BR>\n" );
document.write( "matrices. I was wondering how to solve the problem<BR>\n" );
document.write( "where you have to find the x,y, and z values in the<BR>\n" );
document.write( "matrix:<BR>\n" );
document.write( "[7 -7 5 | 9]<BR>\n" );
document.write( "[9 5 -7 | -17]<BR>\n" );
document.write( "[6 1 -7 | -2]<BR>\n" );
document.write( "I'd appreciate the help. Thank you!<BR>\n" );
document.write( "Caitlyn Reese<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 9969 by venugopalramana(585) About Me on<BR>\n" );
document.write( "2005-11-28 07:14:11 (Show Source):<BR>\n" );
document.write( "the 4 column heads represent x,y,z and constant term<BR>\n" );
document.write( "in the matrix of system of eqns.<BR>\n" );
document.write( "then each row gives us one eqn.like say row 1 gives us<BR>\n" );
document.write( "that 7x-7y+5z=9..etc&#65533;<BR>\n" );
document.write( "hence if we can make the matrix to become<BR>\n" );
document.write( "1 0 0 ?<BR>\n" );
document.write( "0 1 0 ??<BR>\n" );
document.write( "0 0 1 ???<BR>\n" );
document.write( "then from the explantion given above it means<BR>\n" );
document.write( "1x=?.1y=?? And 1z=???<BR>\n" );
document.write( "so we try to transform the matrix in to that form..by<BR>\n" );
document.write( "the following steps.<BR>\n" );
document.write( "in fact using the above explanation,you can see that<BR>\n" );
document.write( "what we do at each step is just<BR>\n" );
document.write( "divide each eqn. with a constant/add/subtract etc<BR>\n" );
document.write( "which does not change the basic<BR>\n" );
document.write( "eqn.for ex. dividing row 1 by 7 means change the given<BR>\n" );
document.write( "eqn.7x-7y+5z=9 to x-y+5z/7=9/7<BR>\n" );
document.write( "legend:- or1 means old row 1..nr1 means new row 1&#65533;r1<BR>\n" );
document.write( "means the existing row 1 please note that no changes<BR>\n" );
document.write( "are made in rows other than those mentioned at each<BR>\n" );
document.write( "step.<BR>\n" );
document.write( "start with given matrix &#65533;<BR>\n" );
document.write( "7...... -7..... 5...... 9<BR>\n" );
document.write( "9...... 5...... -7..... -17<BR>\n" );
document.write( "6...... 1...... -7..... -2<BR>\n" );
document.write( "step 1&#65533;we want to make 1st.row 1st.column as<BR>\n" );
document.write( "1&#65533;.so&#65533;.nr1=or1/7...<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "9...... 5...... -7..... -17<BR>\n" );
document.write( "6...... 1...... -7..... -2<BR>\n" );
document.write( "step 2..we want to make 2nd/3rd.rows,col.1 as<BR>\n" );
document.write( "0...so...nr2=or2-9*r1........nr3=or3-6*r1<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "0... 14..... (-7-9*5/7).... (-17-9*9/7)<BR>\n" );
document.write( "0...... 7...... (-7-6*5/7)..... (-2-6*9/7)<BR>\n" );
document.write( "step 3&#65533;we want to make 2nd.row.2nd.col.as<BR>\n" );
document.write( "1..so..nr2=or2/14<BR>\n" );
document.write( "1...... -1..... 5/7.... 9/7<BR>\n" );
document.write( "0...... 1...... (-94/7)/14..... (-200/7)/14<BR>\n" );
document.write( "0...... 7...... (-7-6*5/7)..... (-2-6*9/7)<BR>\n" );
document.write( "step 4..we want to make 3rd.row.2nd.col.as<BR>\n" );
document.write( "0&#65533;so&#65533;.nr3=or3-7*r2<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "0...... 1...... (-94/7)/14..... (-200/7)/14<BR>\n" );
document.write( "0 0 (-79/7)-7*(-94/98) (68/7)-7*(-200/98)<BR>\n" );
document.write( "step 5&#65533;.we want to make 3rd.row.3rd.col.as<BR>\n" );
document.write( "1&#65533;so&#65533;.nr3=or3/(-32/7)<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "0...... 1...... (-94/98)... (-200/98)<BR>\n" );
document.write( "0...... 0..... 1...... -1<BR>\n" );
document.write( "step 6&#65533;we want to make 1st/2nd.row 3rd.col.as<BR>\n" );
document.write( "0..so..nr1=or1-5*r3/7...nr2=or2+94*r3/98<BR>\n" );
document.write( "1...... -1..... 0..... 2<BR>\n" );
document.write( "0...... 1...... 0...... -3<BR>\n" );
document.write( "0...... 0...... 1...... -1<BR>\n" );
document.write( "step7&#65533;.we want to make 1st.row 2nd.col.as<BR>\n" );
document.write( "0..so&#65533;.nr1=or1+r2<BR>\n" );
document.write( "1...... 0...... 0...... -1<BR>\n" );
document.write( "0...... 1...... 0...... -3<BR>\n" );
document.write( "0...... 0...... 1...... -1<BR>\n" );
document.write( "so x=-1.....y=-3.....and z=-1...you can check back<BR>\n" );
document.write( "YOU CAN SEE THE FOLLOWING ADDITIONAL MATERIAL FOR<BR>\n" );
document.write( "REFERENCE<BR>\n" );
document.write( "How do I perform the next required row operation on<BR>\n" );
document.write( "the following matrix and provide only the next table:<BR>\n" );
document.write( "x y z<BR>\n" );
document.write( "1 28 14 245<BR>\n" );
document.write( "0 3 7 42<BR>\n" );
document.write( "0 7 7 -38<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 9892 by venugopalramana(370) About Me on<BR>\n" );
document.write( "2005-11-25 08:01:32 (Show Source):<BR>\n" );
document.write( "trust you want to solve the equations for x,y and z<BR>\n" );
document.write( "and you are at this stage now....assuming that<BR>\n" );
document.write( ".....our objective is to finally get the matrix if<BR>\n" );
document.write( "possible into the following form ....(i am using<BR>\n" );
document.write( "....to seperate the numbers with suitable gaps..your<BR>\n" );
document.write( "typing is giving raise to uneven gaps bringing a<BR>\n" );
document.write( "little lack of clarity)<BR>\n" );
document.write( "1.....0.....0.....x<BR>\n" );
document.write( "0.....1.....0.....y<BR>\n" );
document.write( "0.....0.....1.....z<BR>\n" );
document.write( "now we have<BR>\n" );
document.write( "1......28.....14.....245<BR>\n" );
document.write( "0.......3......7......42<BR>\n" );
document.write( "0.......7......7.....-38<BR>\n" );
document.write( "new row2=old row2/3.......to get 1 as required in<BR>\n" );
document.write( "row2.so we get...<BR>\n" );
document.write( "1......28.....14.....245<BR>\n" );
document.write( "0......3/3....7/3....42/3<BR>\n" );
document.write( "0.......7......7.....-38<BR>\n" );
document.write( "new row3=oldrow3-7*row2 to get 0 as required in row3<BR>\n" );
document.write( "1......28.....14...........245<BR>\n" );
document.write( "0.......1.....7/3...........14<BR>\n" );
document.write( "0......7-7*1..7-7*7/3......-38-7*14<BR>\n" );
document.write( "new row3 = old row3/(-28/3)..to get 1 as required in<BR>\n" );
document.write( "row3<BR>\n" );
document.write( "1......28.....14...................245<BR>\n" );
document.write( "0.......1.....7/3...................14<BR>\n" );
document.write( "0.......0....(-28/3)/(-28/3).....(-136)/(-28/3)<BR>\n" );
document.write( "this gives us finally in the following form<BR>\n" );
document.write( "1......28.....14............245<BR>\n" );
document.write( "0.......1.....7/3...........14<BR>\n" );
document.write( "0.......0......1............102/7<BR>\n" );
document.write( "now we go back in the same way to get 0 in row2 and<BR>\n" );
document.write( "row3<BR>\n" );
document.write( "new row2=old row2-row3*7/3...and new row1=old<BR>\n" );
document.write( "row1-row3*14...so we get<BR>\n" );
document.write( "1......28......14-1*14.......245-(102/7)*14<BR>\n" );
document.write( "0.......1.......7/3-(7/3)*1...14-(102/7)*(7/3)<BR>\n" );
document.write( "0.......0.........1.............102/7<BR>\n" );
document.write( "the above on simplification gives us<BR>\n" );
document.write( "1.......28.......0..........41<BR>\n" );
document.write( "0........1.......0..........-20<BR>\n" );
document.write( "0........0.......1..........102/7<BR>\n" );
document.write( "now finally we take new row1=old row1-28*row2<BR>\n" );
document.write( "1.......28-28*1......0.......41-(-28*20)<BR>\n" );
document.write( "0........1.......0...........-20<BR>\n" );
document.write( "0........0.......1...........102/7<BR>\n" );
document.write( "so the final answer is<BR>\n" );
document.write( "1......0.......0.......601<BR>\n" );
document.write( "0......1.......0.......-20<BR>\n" );
document.write( "0......0.......1.......102/7<BR>\n" );
document.write( "which tells us that<BR>\n" );
document.write( "1*x+0*y+0*z=x=601<BR>\n" );
document.write( "0*x+1*y+0*z=y=-20<BR>\n" );
document.write( "0*x+0*y+1*z=z=102/7<BR>\n" );
document.write( "note that each and every transformation we did above<BR>\n" );
document.write( "can be interpreted as given in the last statement<BR>\n" );
document.write( "given above...this i hope will give you the insight of<BR>\n" );
document.write( "the process at every step.you can also substitute<BR>\n" );
document.write( "these values of x,y and z in each and every matrix<BR>\n" );
document.write( "above to see that they satify all the equations given<BR>\n" );
document.write( "by the different matrices..in general each mtrix can<BR>\n" );
document.write( "be taken as a set of simltanous equations in x,y and<BR>\n" );
document.write( "z...they can be written as follows..take column 1 is<BR>\n" );
document.write( "for x,column 2 is for y and column 3 is for z.so the<BR>\n" );
document.write( "first matrix you gave<BR>\n" );
document.write( "1......28.....14.....245<BR>\n" );
document.write( "0.......3......7......42<BR>\n" );
document.write( "0.......7......7.....-38<BR>\n" );
document.write( "tells us that<BR>\n" );
document.write( "1*x+28*y+14*z=245....etc...<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );