Cramer's rule - is one of the exact methods of solving systems of linear algebraic equations (Slough).Its accuracy due to the use of determinants of matrices, as well as some of the restrictions imposed in the proof of the theorem.
system of linear algebraic equations with coefficients belonging to, for example, a plurality of R - real numbers, from unknown x1, x2, ..., xn is called the set of expressions of the form
ai2 x1 + ai2 x2 + ... ain xn = bi for i =1, 2, ..., m, (1)
where aij, bi - are real numbers.Each of these expressions is called a linear equation, aij - coefficients of the unknowns, bi - free coefficients of the equations.
solution of (1) is called the n-dimensional vector x ° = (x1 °, x2 °, ..., xn °), which when substituted in for the unknowns x1, x2, ..., xn each of the rows in the system becomestrue equality.
system is called consistent if it has at least one solution, and inconsistent, if its set of solutions coincides with the empty set.
It must be remembered that in order to find the solution of systems of linear algebraic equations using Cramer's rule, matrices, systems must be square, which basically means the same number of unknowns and equations in the system.
So, to use the method of Cramer, you should at least know what the Matrix is a system of linear algebraic equations and how it is issued.And secondly, to understand what is called the determinant of the matrix, and master the skills of its calculation.
assume that this knowledge you possess.Great!Then you have to just memorize formulas determining the method of Cramer.To simplify the memorization use the following notation:
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Det - the main determinant of the system;
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deti - is the determinant of the matrix obtained from the main matrix of the system by replacing the i-th column of the matrix to a column vector whose elements are the right sides of the systems of linear equations;
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n - the number of unknowns and equations in the system.
Then Cramer's rule compute the i-th component xi (i = 1, .. n) n-dimensional vector x can be written as
xi = deti / Det, (2).
Thus Det strictly nonzero.
unique solution when it is jointly provided by the condition of nonzero principal determinant of the system.Otherwise, if the sum of (xi), squared, is strictly positive, then SLAE a square matrix is inconsistent.This can occur in particular when at least one of deti nonzero.
Example 1 .To solve the three-dimensional system of Lau, using Cramer's formula.
x1 + 2 x2 + 4 x3 = 31,
5 x1 + x2 + x3 = 2 29,
3 x1 - x2 + x3 = 10.
decision.We write the matrix of row where Ai - is the i-th row of the matrix.
A1 = (1 2 4), A2 = (1, 5 2), A3 = (-1 3 1).
column free coefficients b = (31 October 29).
main determinant Det system is
Det = a11 a22 a33 + a12 a23 a31 + a31 a21 a32 - a13 a22 a31 - a11 a32 a23 - a33 a21 a12 = 1 - 20 +12 - 12 +2 - 10 = -27.
To calculate det1 use substitution a11 = b1, a21 = b2, a31 = b3.Then
det1 = b1 a22 a33 + a12 a23 b3 + a31 b2 a32 - a13 a22 b3 - b1 a32 a23 - a33 b2 a12 = ... = -81.
Similarly, to calculate a permutation using det2 = b1 a12, a22 = b2, b3 = a32 and, respectively, to calculate det3 - a13 = b1, b2 = a23, a33 = b3.
Then you can check that det2 = -108, and det3 = - 135.
According to Cramer's rule we find x1 = -81 / (-27) = 3, x2 = -108 / (-27) = 4, x3 = -135/ (- 27) = 5.
Answer: x ° = (3,4,5).
Based on the conditions for the applicability of this rule, Cramer's rule for solving systems of linear equations can be used indirectly, for example, to investigate the system on the possible number of solutions depending on the value of a parameter k.
Example 2. Determine for what values of the parameter k the inequality | kx - y - 4 | + | x + ky + 4 | & lt; = 0 has exactly one solution.
decision.
This disparity in the definition of the module function can be performed only if both expressions are zero simultaneously.Therefore, this problem is reduced to finding the solution of a linear system of algebraic equations
kx - y = 4,
x + ky = -4.
solution of this system only if it is the main determinant of
Det = k ^ {2} + 1 is nonzero.Obviously, this condition holds for all valid values of the parameter k.
Answer: for all real values of the parameter k.
The objectives of this type can also be reduced, many practical problems of mathematics, physics or chemistry.
