Cramer's Rule
The
method of solution of
linear equations by determinants is called Cramer's Rule.
This rule for linear equations in 3 unknowns is a method of solving by
determinants the following equations for x,
y, z
a_{1}x + b_{1}y + c_{1}z
= d_{1}
a_{2}x + b_{2}y + c_{2}z = d_{2}
a_{3}x + b_{3}y + c_{3}z = d_{3}_{ }
If we
analytically
solve
the equations above, we obtain
If 

is the determinant of coefficients
of x, y,
z
and is 
assumed
not equal to zero, then we may rewrite the values as
The solution
involving determinants is easy to remember if you keep in mind these
simple
ideas:
 The denominators are given by
the determinant in which the elements are the coefficients of x, y
and z, arranged as in the original
given equations.
 The numerator in the solution
for any variable is the same as the determinant of the coefficients ∆ with the exception that the
column of coefficients of the unknown to be determined is replaced by
the column of constants on the right side of the original
equations.
That is,
for the first variable, you substitute the first column of the
determinant with the constants on the right; for the second
variable, you substitute the second column with the constants
on the rigth, and so on...
Example:
Solve
this system using Cramer’s Rule
2x + 4y
– 2z = 6
6x + 2y
+ 2z = 8
2x – 2y
+ 4z = 12
For x,
take the determinant above and
replace the first column by the constants on the right of the system.
Then,
divide this by the determinant:
For y,
replace the second column by the
constants on the right of the system. Then, divide it by the
determinant:
For z,
replace the third column by the
constants on the right of the system. Then, divide it by the
determinant:
You
just solved ths system!
In
Matlab, it’s even
easier. You can solve the system with just one instruction.
Let D
be the matrix
of just the coefficients of the variables:
D =
[2 4 2;
6
2 2;
2 2
4];
Let b
be the column vector
of the constants on the rigth of the system :
b = [6
8 12]'; % the apostrophe is used to transpose
a vector
Find the
column vector of the unknowns by 'left
dividing' D
by b (use
the backslash), like this:
variables
= D\b
And
Matlab response is:
variables
=
1.0000
1.0000
2.0000
From 'Cramers Rule' to home
From 'Cramers Rule' to 'Linear
Algebra'
