Matrix "hence" question?

Question

Basically I have this matrix question.

I succeeded in the first part, and the answers match.

Here are the answers:

x= (1/22)(a-3b+11c),

y=(1/22)(5a+7b-11c)

z=(1/22)(-13a-5b+11c)

Now the problem really arises from the Hence part.

It also says "otherwise", so I simply found the inverse using the Adjoint method.

However I am curious, what did I miss?

How can I use the results of x y and z to find the inverse?

I do realize that the equations can form the given matrix.

Thanks in advance!!

Answer 1

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You can write the system of equations as M[x,y,z] = [a,b,c] where M is the matrix of coefficients

The solution to this system is [x,y,z] = M⁻¹[a,b,c]

Hence if you know the solution for x,y,z and you express it in form [x,y,z]=N[a,b,c] then N=M⁻¹

Given solutions can be written [x,y,z] = (1/22){ {1,-3,11}, {5,7,-11}, {-13,-5,11} }[a,b,c] (matrix by row)

Hence M⁻¹ = (1/22){ {1,-3,11}, {5,7,-11}, {-13,-5,11} }