 Solving system of three equations in three unknowns using Cramer's rule Method of expanding a determinant of a rank n by cofactors
Cramer’s rule (using the determinant) to solve systems of linear equations
Solving system of two equations in two unknowns using Cramer's rule
A system of two equations in two unknowns, the solution to a system by Cramer’s rule (use of determinants). the solution to the system Example:  Solve given system of linear equations using Cramer’s rule.
 Solution: Solving system of three equations in three unknowns using Cramer's rule A determinant of rank n can be evaluated by expanding to its cofactors of rank n - 1, along any row or column taking into account the scheme of the signs. For example, the determinant of rank n = 3, Example:  Solve given system of three equations in three unknowns using method of expanding to cofactors.
 Solution: Method of expanding a determinant of a rank n by cofactors, example
The value of a determinant will not change by adding multiples of any column or row to any other column or row. This way created are zero entries that simplify subsequent calculations.
Example:  An application of the method of expanding a determinant to cofactors to evaluate the determinant of the rank four. Added is third to the second colon. Then, the second row multiplied by -3 is added to the first row. The obtained determinant is then expanded to its cofactors along the second colon: The first colon multiplied by -1 is added to the third colon. The obtained determinant is then expanded along the third colon.   Matrices, determinants and systems of linear equations contents 