Gauss Elimination in Matrix Form

The Gauss elimination procedure, which was described above in formula form, can also be accomplished by series of matrix multiplications. Two types of special matrices are involved in this operation. Both of these matrices are modifications of the identity matrix. The first type, which we designate as P,, is the identity matrix with the following changes The unity at position ii switches places with the zero at position ij, and the unity at position jj switches places with the zero at position ji. For example, for a fifth-order system is 

Premultiplication of matrix A by Pg has the effect of interchanging rows i and j. Postmultiplication causes interchange of column / and j. By definition, Pg = /, and multiplication of A by causes no interchanges. The inverse of Pg is identical to Py. 

The second type of matrices used by the Gauss elimination method are unit lower triangular matrices of the form 

Therefore, the entire Gauss elimination method, which reduces a nonsingular matrix A to an upper triangular matrix U, can be represented by the following series of matrix multiplications  

The matrices are unit lower triangular and their product, defined by matrix L, is also unit lower triangular. With this definition of L, Eq. (2.72) condenses to 

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Method of Solution The function is written based on Gauss elimination in matrix form. It applies complete pivoting strategy by searching rows and columns for the maximum pivot element. It keeps track of column interchanges, which affect the positions of the unknown variables. The function applies the back-substimtion formula [Eq. (2.110)] to calculate the unknown variables and interchanges their order to correct for column pivoting. 

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