(where
the determinant of 
is not equal
to zero), having unknown variables x1, x2, …xn,
the unique solution of the system can be found by Cramer's Rule
Note: The solution theory in this section requires knowledge of cofactor expansion.
Given a system of linear algebraic equations, (where
the determinant of is not equal
to zero), having unknown variables x1, x2, …xn,
the unique solution of the system can be found by
the matrix Aj is found by substituting the column of known constraints
into the jth column of the matrix .
Example
The same solution was obtained in the Gaussian elimination lesson. One can verify the solution by checking substituting x1, x2, and x3 into A x = b checking if the right-hand side (b) is obtained.