Linear Algebra-Cramer's rule

Cramer's rule is a way of solving for a system of equations assuming that there are n unknowns with n linear equations. This is another way of saying that the coefficient matrix is invertible. Instead of using row reduction methods, Cramer's rule utilizes determinants. Assuming a given coefficient matrix and a solution vector, we first take the determinant of the coefficient matrix. We can then replace the coefficients of a variable x _n with the solution vector and take the determinant of that matrix. If you divide the determinant of the coefficient matrix with the solution vector over the determinant of the coefficient matrix we get the solution to x _n given that the determinant of the coefficient matrix is nonzero.

Example-
|2 3 7  |
|4 5 13|

So to solve this we take the determinant of the coefficient matrix
|2 3|
|4 5|
det=(2)5-3(4)
det=10-12
det=-2

|7   3|
|13 5|
det=(7)5-3(13)
det=35-39
det=-4

|2 7  |
|4 13|
det=(2)13-4(7)
det=26-28
det=-2

x₁ = -2/-2 = 1
x₂ = -4/-2 = 2