When solving a system of linear equations, you can use substitution, elimination to solve. But you also set up the system with matrices and solve using Cramer's rule.

For example,  the system  2x  + 3y = 10
                                       -3x - 6y = 22

Can be solved as follows:

matrix A =  2  3
                 -3  -6

Get the determinant of A = (2)(-6) - (-3)(3) = -3

Now to solve for x, set up another matrix, substituting 10  for  2  in the matrix

                                                                                  22       -3

the new matrix looks like  10 3
                                        22 -3

Get the determinant (10)(-3) - (22)(3) = -30 - 66 = -86

x is the determinant x divided by determinant A = -86/-3   = 86/3

Now do the same thing to get y

the new matrix is     2 10
                              -3 22

Determinant is (2)(22) - (-3)(10) = 44 + 30 = 74

y is 74/-3 = -74/3