# Basic concepts Kramer's method

 Sana 26.05.2018 Hajmi 445 b. • 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 • ## If free members of all equations of system are equal to zero, then the system is called uniform, otherwise – non-uniform. • ## Theorem (Kramer's rule). If determinant of system ∆≠ 0, then the considered system (1) has one and only one decision, and • ## We will make and will calculate necessary determinants: • ## We find unknown on Kramer's formulas: • ## 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: • ## 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. • ## 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: • ## Anser: • ## So we have received the decision of system of three linear equations with three unknown by a matrix method. • ## 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: • ## where • ## It was necessary to calculate a matrix of unknown variables, having increased the return matrix by a matrix column of free members: 