
Systems of Linear
Equations 
Cramer’s
rule (using the determinant) to solve systems of linear equations

Solving system
of two equations in two unknowns using Cramer's rule

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
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