Question 1196243
<pre>
The cofactor method of finding the inverse is the hardest way.
Here goes:

{{{M}}}{{{""=""}}}{{{(matrix(3,3,1,-1,k,4,7,3,-1,12,-2))}}}

{{{"det(M)"}}}{{{""=""}}}{{{abs(matrix(3,3,1,-1,k,4,7,3,-1,12,-2))}}}{{{""=""}}}{{{(1)abs(matrix(2,2,7,3,12,-2))-(-1)abs(matrix(2,2,4,3,-1,-2))+(k)abs(matrix(2,2,4,7,-1,12))}}}{{{""=""}}}

{{{abs(matrix(2,2,7,3,12,-2))+abs(matrix(2,2,4,3,-1,-2))+k*abs(matrix(2,2,4,7,-1,12))}}}{{{""=""}}}

{{{((7)(-2)-(3)(12)^"")+((4)(-2)-(3)(-1)^"")+k((4)(12)-(7)(-1)^"")}}}{{{""=""}}}
{{{(-14-36)+(-8+3)+k(48+7)}}}{{{""=""}}}{{{-50-5+k(55)}}}{{{""=""}}}{{{55k-55}}}

In order to solve the equation 

{{{MX}}}{{{""=""}}}{{{(matrix(3,1,1,11,21))}}}

We must find the inverse of M which is written M<sup>-1</sup> by the long,
hard, cofactor method:

We replace each element of M by the determinant of its cofactor, which is its
minor 2x2 determinant.

{{{(matrix(3,3,

abs(matrix(2,2,7,3,12,-2)), abs(matrix(2,2,4,3,-1,-2)),abs(matrix(2,2,4,7,-1,12)),
abs(matrix(2,2,-1,2,12,-2)),abs(matrix(2,2,1,2,-1,-2)),abs(matrix(2,2,1,-1,-1,12)),
abs(matrix(2,2,-1,2,7,3)),abs(matrix(2,2,1,2,4,3)),abs(matrix(2,2,1,-1,4,7))) )}}}

We put in the "checkerboard" of signs, this will be the cofactor matrix

{{{(matrix(3,3,

""+abs(matrix(2,2,7,3,12,-2)), ""-abs(matrix(2,2,4,3,-1,-2)),""+abs(matrix(2,2,4,7,-1,12)),
""-abs(matrix(2,2,-1,2,12,-2)),""+abs(matrix(2,2,1,2,-1,-2)),""-abs(matrix(2,2,1,-1,-1,12)),
""+abs(matrix(2,2,-1,2,7,3)),""-abs(matrix(2,2,1,2,4,3)),""+abs(matrix(2,2,1,-1,4,7))) )}}}

We evaluate the determinants

{{{(matrix(3,3,

""+((7)(-2)-(3)(12)^""),""-((4)(-2)-(3)(-1)^""),""+((4)(12)-(7)(-1)^""),
""-((-1)(-2)-(2)(12)^""),""+((1)(-2)-(2)(-1)^""),""-((1)(12)-(-1)(-1)^""),
""+((-1)(3)-(2)(7)^""),""-((1)(3)-(2)(4)^""),""+((1)(7)-(-1)(4)^"")))}}}

{{{(matrix(3,3,

""+((-14)-(36)^""),""-((-8)-(-3)^""),""+((48)-(-7)^""),
""-((2)-(24)^""),""+((-2)-(-2)^""),""-((12)-(1)^""),
""+((-3)-(14)^""),""-((3)-(8)^""),""+((7)-(-4)^"")))}}}

{{{(matrix(3,3,

""+(-14-36^""),""-(-8+3^""),""+(48+7^""),
""-(2-24^""),""+(-2+2^""),""-(12-1^""),
""+(-3-14^""),""-(3-8^""),""+(7+4^"")))}}}


{{{(matrix(3,3,

""+(-50^""),""-(-5^""),""+(55^""),
""-(-22^""),""+(0^""),""-(11^""),
""+(-17^""),""-(-5^""),""+(11^"")))}}}

{{{(matrix(3,3,

-50,5,55,
22,0,-11,
-17,5,11))}}}

Next we form the adjoint or adjugate matrix which is the transpose
of the matrix of cofactors:

{{{(matrix(3,3,

-50,22,-17,
5,0,5,
55,-11,11))}}}

Now we find the inverse of M by multiplying the adjoint or adjugate
by the reciprocal of the value of the determinant of M. We calculated
the value of the determinant of M in terms of k as
{{{55k-55}}}
But we were told to use k=2, so substituting,
{{{55(2)-55}}}{{{""=""}}}{{{110-55}}}{{{""=""}}}{{{55}}}
So we multiply the adjoint or adjugate by {{{1/55}}}


{{{M^(-1)}}}{{{""=""}}}{{{expr(1/55)(matrix(3,3,

-50,22,-17,
5,0,5,
55,-11,11))}}}{{{""=""}}}{{{(matrix(3,3,

-50/55,22/55,-17/55,
5/55,0/55,5/55,
55/55,-11/55,11/55))}}}{{{""=""}}}{{{(matrix(3,3,

-10/11,2/5,-17/55,
1/11,0,1/11,
1,-1/5,1/5))}}}


Next we multiply both sides of the equation we are to solve:

{{{MX}}}{{{""=""}}}{{{(matrix(3,1,1,11,21))}}}

by the inverse M<sup>-1</sup>:

{{{M^(-1)(MX^"")}}}{{{""=""}}}{{{(M^(-1)M)X}}}{{{""=""}}}{{{IX}}}{{{""=""}}}{{{X}}}{{{""=""}}}{{{(matrix(3,1,x,y,z))}}}{{{""=""}}}{{{(matrix(3,3,

-10/11,2/5,-17/55,
1/11,0,1/11,
1,-1/5,1/5))(matrix(3,1,1,11,21)))}}}{{{""=""}}}{{{

(matrix(3,1,

(-10/11)(1)+(2/5)(11)+(-17/55)(21),
(1/11)(1)+(0)(11)+(1/11)(21),
(1)(1)+(-1/5)(11)+(1/5)(21)))}}}{{{""=""}}}

{{{(matrix(3,1,

-10/11+22/5-357/55,
1/11+0+21/11,
1-11/5+21/5))}}}{{{""=""}}}{{{(matrix(3,1,

-50/55+242/55-357/55,
1/11+21/11,
5/5-11/5+21/5))}}}{{{""=""}}}{{{(matrix(3,1,

-165/55,
22/11,
15/5))}}}{{{""=""}}}{{{(matrix(3,1,-3,2,3))}}}

So

{{{X}}}{{{""=""}}}{{{(matrix(3,1,x,y,z))}}}{{{""=""}}}{{{(matrix(3,1,-3,2,3))}}}

Edwin</pre>