Cramer's Rule
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ScribdTranslationsCramer's Rule
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ScribdTranslationsCramer's rule is atheoremin linear algebra,that provides the solution to asystem
linear equationsin terms ofdeterminants.Receives this name in honor of
Gabriel Cramer(1704 - 1752), who published the rule in his Introduction to the Analysis of
algebraic curves of 1750, althoughColin Maclaurinhe also published the method
in his Treatise of Geometry from 1748 (and he probably knew of the method since 1729).[1]
Cramer's rule is of theoretical importance because it provides an explicit expression for the
solution of the system. However, for systems of linear equations of more than three
equations, their application for solving it is excessively costly:
computationally, it is inefficient for large matrices and therefore it is not used in
practical applications that may involve many equations. However, as not
it is necessary to pivot matrices, it is more efficient than theGaussian eliminationfor
small matrices, particularly when they are used for operationss SIMD.
Yes it is a system of equations. it is the coefficient matrix of the system,
it is the column vector of the unknowns and it is the column vector of
the independent terms. Then the solution to the system is presented as follows:
where it is the resulting matrix from replacing the j-th column of by the vector
column. It should be noted that for the system to be compatible, a certain
determinant of the matrix it must be non-null.
Explicit formulas for small systems
System of 2 equations with 2 unknowns
For the resolution of a system of two equations with two unknowns, of the form.
Given the system of equations:
We represent it in the form of matrices:
Then, x and y can be found using Cramer's rule, with a division of
determinantsin the following way:
System of 3 equations with 3 unknowns
The rule for a system of three equations with three unknowns is similar, with a
division ofand determinants:
Which represented in the form of a matrix is:
x, y, z can be found as follows:
Demonstration
Sean:
Using the properties of lmatrix multiplication:
So:
Therefore:
Besides, remembering the definition ofdeterminantthe defined summation accumulates to
multiplication of the adjoint element or cofactor of position ij, with the i-th element
of the vector (which is precisely the i-th element of column j, in the matrix ).
Cramer obtained the unknowns solved from a system in terms of determinants.
Let's solve the system:
The formulas are:
Let's remember that the formula for determinants (3x3) is:
As can be seen, in order to use Cramer's method, the
the determinant of the coefficient matrix must not be 0 for the denominator of
the formulas are not null. If it gives 0, it means that one of the unknowns can be set to
function of the others, that is, we would have parameters. The way to solve this
the problem is to move the unknown to the other member (next to the constant term) that
let's take as a parameter and in this way we will have a determinant that does not become null
but of a lower degree. When applying Cramer's formula we will have a parameter in the
column of the independent terms.
Cramer's method is used to solve systems of linear equations. It is applied to
systems that meet the following two conditions:
The number of equations is equal to the number of unknowns.
The determinant of the coefficient matrix is different from zero.
Such systems are called Cramer systems.
Let Δ be the determinant of the coefficient matrix.
And they are:
Δ1, Δ2, Δ3 ... , Δn
the determinants obtained by substituting the coefficients of the 2nd member (the terms
independents) in the 1st column, in the 2nd column, in the 3rd column and in the nth
column respectively.
A Cramer system has a single solution given by the following
expressions
Exercises: