Question 442543
Given a matrix:

{{{matrix(2,2,2,5,3,1)}}}

Cramer's rule states that the xi = det(Ai)/det(A)

So what is the det(A)?  det(A) = 2*1 - 3*5 = -13

You do not specify the solution matrix in this problem, but I will act like you did and you can apply it to your problem.

Let's say the solution matrix is {{{matrix(2,1,1,2)}}}.

Then to solve for x1 you would create a matrix where the solution matrix is substituted in for column 1.

In this made up example, you'd find the det(A1) where A1 is:

{{{matrix(2,2,1,5,2,1)}}}

This determinant is 1 -10 = -9.

So the solution to x1 = -9/-13 = (9/13)

To find x2 create an A2 matrix.  

The matrix would be:

{{{matrix(2,2,2,1,3,2)}}}

The det(A2) = 4 - 3 = 1

So then x2 = 1 / -13 =  (-1/13)

To show that Cramer's rule indeed works:

2x1 + 5x2 = 1
3x1 + 1x2 = 2

Where as I said {{{matrix(1,2,1,2)}}} is my solution matrix and {{{matrix(2,2,2,5,3,1)}}} was the matrix you gave me.

If I plug in x1=9/13 and x2=-1/13

2(9/13) + 5(-1/13) = 13/13 = 1.  Check.
3(9/13) + 1(-1/13) = 26/13 = 2.  Check.

Now for whatever your solution matrix is, apply the same steps.