Lab for the Gauss-Jordan Method

Module for the Gauss-Jordan Method

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Gauss-Jordan Method for Linear Systems. Construct the solution to AX = B, by using Gaussian elimination. Let the coefficients of A and the constants of B be stored in the augmented matrix
[A, B] = () which has n rows and n+1 columns; that is, the column vector B = () is
stored in column n+1 of the augmented matrix [A, B]. The system can be written

Row operations will be used to eliminate in column p.
Remark. In the following subroutine the notation i=!=p means "i not equal to j".

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Report to be handed in.

Computer Problems.

Exercise 1. Use above Gaussian elimination method to solve the linear system AX = B:

First form the augmented matrix M = [A,B]

Then perform Gaussian elimination.
Remark. All the computations are printed too !

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Verify the solution.

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Exercise 2. Form the augmented matrix M using the following steps.
Go the Input menu and pull down the Create/Table/Matrix/Palette and form a 4 by 5 matrix.

Then store A in the first four columns of M.

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Then store B in the fifth column of M.
You will need to Flatten B when you do this !

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Then perform Gaussian elimination as you did before, showing all the intermediate computations.

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This time get the solution vector X out of this augmented matrix !

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What did you just learn about the data structure of a row vector vs the data structure of a column vector ?

Now use the subroutine Gauss for finding the inverse of a matrix.

Exercise 3. Use above Gaussian elimination method to find the inverse of the matrix A.

Form the augmented matrix M = [A, I] using the following steps.
Remark. Don't use I as a variable, it is a reserved word which is the complex constant .
Go the Input menu and pull down the Create/Table/Matrix/Palette and form a 4 by 8 matrix.

Go the Input menu and pull down the Create/Table/Matrix/Palette and create a 4 by 4 identity matrix.

Then store A in the first four columns of M.

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Then store Iden in the last four columns of M.

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Then perform Gaussian elimination.

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Get the inverse of A out of this augmented matrix ! And store it in the matrix B.

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Verify that B is the inverse of A.

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(c) John H. Mathews, 1998