tailieunhanh - Handbook of mathematics for engineers and scienteists part 34

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 34', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Systems of Linear Algebraic Equations 199 . General square system of linear equations. A square system of linear equations has the form AX B where A is a square matrix. 1 . If the determinant of system is different from zero . det A 0 then the system has a unique solution X A B. 2 . Cramer rule. If the determinant of the matrix of system is different from zero . A det A 0 then the system admits a unique solution which is expressed by A1 A2 An x1 a x2 a xn a where Ak k 1 2 . n is the determinant of the matrix obtained from A by replacing its kth column with the column of free terms a11 a12 b1 a1n A .- a21 . . a22 . . b2 . . . a2n . . . . an1 . an2 . bn . ann Example 1. Using the Cramer rule let us find the solution of the system of linear equations 2x x2 4x3 16 3x1 2x2 x3 10 x1 3x2 3x3 16. The determinant of its basic matrix is different from zero 2 1 4 and we have 16 1 4 A1 10 2 1 26 16 3 3 A 3 2 1 26 0 1 3 3 2 16 4 2 1 16 A2 3 10 1 52 A3 3 2 10 78. 1 16 3 1 3 16 Therefore by the Cramer rule the only solution of the system has the form A1 26 A2 52 X1 T 26 1 X2 T 26 2 78 3. 3 . Gaussian elimination of unknown quantities. Two systems are said to be equivalent if their sets of solutions coincide. The method of Gaussian elimination consists in the reduction of a given system to an equivalent system with an upper triangular basic matrix. The latter system can be easily solved. This reduction is carried out in finitely many steps. On every step one performs an elementary transformation of the system or the corresponding augmented matrix and obtains an equivalent system. The elementary transformations are of the following three types 1. Interchange of two equations or the corresponding rows of the augmented matrix . 2. Multiplication of both sides of one equation or the corresponding row of the augmented matrix by a nonzero constant. 3. Adding to both sides of one equation both sides of another equation multiplied