By S. K. Singh
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23) can be rewritten with the values just found LM2 A = M3 MN3 OP P 5PQ LM MM N3 −1 1 2 0 3 9 = 3 9/ 2 3 9/ 2 OP LM1 PP MM0 −4Q N0 0 0 OP P 1 PQ −1/ 2 1/ 2 1 5/ 3 = LU 0 Now, selecting an equation LZ = b, where Z is a 3 × 1 column matrix, yields LM2 MM3 N3 0 9/ 2 9/ 2 OP LM z OP LM−1OP PP MMz PP = MM 0 PP −4Q N z Q N 4 Q 0 0 1 2 3 that provides the determined values for z1, z2 and z3. LM z OP LM−1/ 2OP MMz PP = MM 1/ 3 PP . Nz Q N −1 Q 1 Therefore 2 3 Now on account of the relation Ux = Z we obtain for the solution to linear system.
The program g3GAUELI implements the computational technique for the solution of the system. Cramer’s rule, the most fundamental method for direct computation of the solution, has been applied using the techniques of determinants. 2. The method of Matrix Inversion, that I have described earlier in this section, is used to compute the system of equations when the system is represented in matrix form Ax = b. The solution of the linear system becomes x = A–1b, if we can find out the inverse matrix, A–1.
Now we can define a diagonal matrix when the entries or elements of A other than those down the leading or principal diagonal are zero. In other words, a diagonal matrix D is in this case a square matrix of order 4 with off diagonal elements being zero and is represented by LMa 0 D=M MM 0 N0 0 11 a22 0 0 0 0 a33 0 OP P. 0 P P a Q 0 0 44 The identity matrix of order 4 is a diagonal matrix of order 4, in which all the diagonal entries equal to 1 and all other off-diagonal entries equal to 0. Hence LM1 0 D=M MM0 N0 0 1 0 0 0 0 1 0 OP PP PQ 0 0 .
Engineering Chemistry by S. K. Singh