Reduced Echelon Form of a Matrix

Linear Algebra Project

Objective:

To reduce a matrix to its echelon form by elementary row operations and use it to solve systems of linear equations. While doing this you will begin to learn Maple's linear algebra features.

Computer will do the lengthy arithmetic operations for you. This will help you to utilize your time and energy on understanding the reduced echelon form of a matrix and the process for attaining it. You will find this very useful later on when you study the structure and transformations of a vector space.

Solved Example:

Solve the following system of equations using elementary row operations to reduce its augmented matrix to echelon form.

Solution:

We activate the linear algebra package.

> with(linalg);

First let us write the augmented matrix of the system:

> M:=matrix(3,4,[[2,6,-1,8],[3,9,0,15],[-2,-5,6,1]]);

Now we proceed to construct the row-reduced echelon matrix.

Multiply the first row of the matrix by to obtain matrix M1.

> M1:= mulrow(M,1,1/2);

> M2:=pivot(M1,1,1);

Do you see what pivot operation has done?

Pivot: 1 in the first row and first column of the matrix M1 is used as a lever to turn the rest of the entries in column one into 0. To do this the first row of M1 is multiplied by -3 and added to the second row, and the first row is multiplied by 2 and added to the third row. The resulting matrix is M2.

Matrices M1 and M2 are "equivalent", meaning that the systems of equations that they represent have exactly the same set of solutions.

Now swap rows 2 and 3 of matrix M2 to get matrix M3.

> M3:= swaprow(M2,2,3);

> M4:=pivot(M3,2,2);

> M5:=mulrow(M4,3,2/3);

> M6:=pivot(M5,3,3);

M6 is the row reduced echelon form of the augmented matrix M. This yields the solution to the given system as , and .

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ASSIGNMENT

Problem 1:

Use Maple as we did above to reduce the augmented matrix of the following system of equations to its echelon form. Discuss the nature of solutions and find a solution set for the system.

Problem 2:

Use Maple to reduce the augmented matrix of the following system of equations to its echelon form. Discuss the nature of solutions and find a solution set for the system, if possible.

Problem 3:

Does the following system have a solution?

Problem 4:

Maple has a procedure that automatically computes the row-reduced echelon form of a matrix, it is denoted by "gaussjord". As an example let us use it on the matrix M which is the augmented matrix of the system of equations in the "solved example" we had above.

> M:=matrix(3,4,[[2,6,-1,8],[3,9,0,15],[-2,-5,6,1]]);

> gaussjord(M);

You see that this is exactly the same row-reduced echelon matrix that we got in the solved example.

Now apply the "gaussjord" command to the augmented matrices you got in problems 1 and 2. Do you get the same echelon matrices as you did before?

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MSIP Grant #120A80089-98: "Three Urban Calculus Reform Programs: Adopting the Best" 1998-2001