Basic concepts Kramer's method


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    • Basic concepts
    • Kramer's method
    • Decision of system Kramer's by method
    • Gauss method
    • Decision of system by the Gauss method
    • Matrix method (by means of the return matrix)
    • Decision of system by a matrix method


We will consider system of three linear equations with three unknown:

  • We will consider system of three linear equations with three unknown:

  • Where- -unkown , - coefficients ( ),

  • -- free members.

  • The three of numbers is called the decision of system of three linear equations with three unknown if at substitution they in the equations of system instead of are received by right numerical equalities.

  • If the system of three linear equations has at least one decision, then it is called joint.

  • If the system of three linear equations of decisions has no, then it is called not joint.

  • If the system of three linear equations has the only decision, then it call certain; if decisions more than one, then – uncertain.

  • If free members of all equations of system are equal to zero, then the system is called uniform, otherwise – non-uniform.



Let us need to solve system of three linear equations with three unknown:

  • Let us need to solve system of three linear equations with three unknown:

  • (1)

  • in which determinant of system (it is made of coefficients at unknown) ∆≠ 0, and determinants turn out from determinant of system ∆ by means of replacement with free members of elements according to the first, second and third columns.

  • Theorem (Kramer's rule). If determinant of system ∆≠ 0, then the considered system (1) has one and only one decision, and



Decision:

  • Decision:

  • We will calculate determinant of system:

  • As the determinant of system is other than zero, the system has the only decision which can be found Kramer's method.

  • We will make and will calculate necessary determinants:



We find unknown on Kramer's formulas:

  • We find unknown on Kramer's formulas:

  • Answer:



Earlier considered method can be applied at the decision only of those systems in which the number of the equations coincides with number of unknown, and the determinant of system has to be other than zero. The method of Gauss is more universal and is suitable for systems with any number of the equations. It consists in a consecutive exception of unknown of the equations of system.

  • Earlier considered method can be applied at the decision only of those systems in which the number of the equations coincides with number of unknown, and the determinant of system has to be other than zero. The method of Gauss is more universal and is suitable for systems with any number of the equations. It consists in a consecutive exception of unknown of the equations of system.

  • Again we will consider system of three linear equations with three unknown:

  • We will leave the first equation without change, and we will exclude the composed, containing x1 from the 2nd and 3rd. For this purpose we will divide the second equation on a21 and we will increase on – a11, and then we will put with the 1st equation. Similar to the third equation we will divide on a31 and we will increase on – a11, and then we will put with the first. As a result the initial system will take a form:



Now we will exclude the composed, containing x2 from the last equation. For this purpose we will divide the third equation on, we will increase on and we will put with the second. We will have then system of the equations:

  • Now we will exclude the composed, containing x2 from the last equation. For this purpose we will divide the third equation on, we will increase on and we will put with the second. We will have then system of the equations:

  • From here from the last equation it is easy to find x3, then from the 2nd equation of x2 and, at last, from the 1st – x1.



Решение:

  • Решение:

  • We will leave the first equation without change, and we will exclude the composed, containing x1 from the 2nd and 3rd. For this purpose we will increase the second equation on, and then we will put with the 1st equation.

  • Similar to the third equation we will increase on, and then we will put with the first.

  • As a result the initial system will take a form:

  • Now we will exclude the composed, containing x2 from the last equation. For this purpose we will increase the third equation on, and we will put with the second. We will have then system of the equations:



On it the forward stroke of a method of Gauss is finished, we begin reverse motion.

  • On it the forward stroke of a method of Gauss is finished, we begin reverse motion.

  • From the last equation of the received system of the equations we find x3:

  • From the second equation we receive:

  • From the first equation we find the remained unknown variable and it has finished reverse motion of a method of Gauss:

  • Anser:



We will consider system of three linear equations with three unknown:

  • We will consider system of three linear equations with three unknown:

  • In a matrix form of record this system of the equations has an appearance where

  • Let . Then there is the return matrix . If to increase both parts of equality on at the left ,then we will receive a formula for finding of a matrix column of unknown variables, i.e.

  • or.

  • So we have received the decision of system of three linear equations with three unknown by a matrix method.



Decision:

  • Decision:

  • We will rewrite system of the equations in a matrix form:

  •  Just as

  • that system of three linear equations with three unknown can be solved by a matrix method. By means of the return matrix the solution of this system can be found as:



We will construct the return matrix by means of a matrix of algebraic additions of elements of a matrix :

  • We will construct the return matrix by means of a matrix of algebraic additions of elements of a matrix :

  • where



It was necessary to calculate a matrix of unknown variables, having increased the return matrix by a matrix column of free members:

  • It was necessary to calculate a matrix of unknown variables, having increased the return matrix by a matrix column of free members:

  • Answer: .




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