Question 136905
Step 1: Create the coefficient determinant and the answer column matrix


{{{D=(matrix(3,3,3,-2,1,4,-4,3,5,-4,1))}}}
{{{AC=(matrix(3,1,6,0,-5))}}}


Step 2: Evaluate the determinant.


{{{cartoon(
(matrix(3,5,red(a),b,c,a,b,d,red(e),f,d,e,g,h,red(i),g,h))=red(aei),
(matrix(3,5,a,red(b),c,a,b,d,e,red(f),d,e,g,h,i,red(g),h))=aei+red(bfg),
(matrix(3,5,a,b,red(c),a,b,d,e,f,red(d),e,g,h,i,g,red(h)))=aei+bfg+red(cdh),(matrix(3,5,a,b,red(c),a,b,d,red(e),f,d,e,red(g),h,i,g,h))=aei+bfg+cdh-(red(gec)),
(matrix(3,5,a,b,c,red(a),b,d,e,red(f),d,e,g,red(h),i,g,h))=aei+bfg+cdh-(gec+red(hfa)),
(matrix(3,5,a,b,c,a,red(b),d,e,f,red(d),e,g,h,red(i),g,h))=aei+bfg+cdh-(gec+hfa+red(idb)))}}}


{{{green(D)=aei+bfg+cdh-gec-hfa-idb=red((3*-4*1)+(-2*3*5)+(1*4*-4)-(5*-4*1)-(-4*3*3)-(1*4*-2))=-12-30-16+20+36+8=green(6)}}}


Step 3: Now, take the coefficient determinant and replace the x-column (the first column) with the answer column, like this:


{{{D[x]=(matrix(3,3,6,-2,1,0,-4,3,-5,-4,1))}}}


Step 4: Evaluate the {{{D[x]}}} determinant as shown in step 2.


Cramer's rule states that {{{x=D[x]/D}}}


Step 5: Repeat step 3 and 4, except replace the y-column (the second column) with the answer column matrix and apply Cramer's rule: {{{y=D[y]/D}}}


Step 6: Do it one more time replacing the z-column (the third column) with the answer column.  {{{z=D[z]/D}}}


Hint: Be extremely meticulous about the signs on the numbers when evaluating the determinants.  If you have access to Microsoft Excel, you can put the coefficients into a square set of cells, one coefficient per cell.  Let's say you put them in cells A1 through C3, then in an empty cell, type in =MDETERM(A1:C3).  When you hit enter, the value of the determinant will appear.