Tutorial Math 204 Tutorial 1 System of linear equations Gauss / Gauss - Jordan reduction

1. Problem 1

1.1. Find the augmented matrix of the system

1.2. Gauss/Gauss Jordan Reduction

1.2.1. There are three types of elementary row operations which may be performed on the rows of a matrix: Type 1: Swap the positions of two rows. Type 2: Multiply a row by a nonzero scalar. Type 3: Add to one row a scalar multiple of another.

1.3. The system with zeros

1.3.1. The Augmented Matrix is:

1.3.1.1. Now we use the algorithm

1.3.1.1.1. Second operation

1.4. Reduce echalon form

1.4.1. What is it?

1.4.2. So…? No equation of this system has a form zero = nonzero; Therefore, the system is consistent. The leading entries in the matrix have been highlighted in yellow. A leading entry on the (i,j) position indicates that the j-th unknown will be determined using the i-th equation.

1.4.3. Then… Those columns in the coefficient part of the matrix that do not contain leading entries, correspond to unknowns that will be arbitrary. The system has infinitely many solutions:

1.5. Is this the best way to solve it?

1.5.1. For a computer YES (almost)

1.5.2. For Humans NO… Let's try to solve it easier.

1.5.2.1. Go to this site:

2. Probelm 2 to be solved in the blackboard

3. Problem 3 to be solved in the Blackboard by students

4. Problem 4

5. Important Points To Remark

5.1. How to write the steps in the Aug.Coeff.Matrix(ACM). Write the ACM and beside it state the Elementary Row Operation(ERO) to be executed.

5.2. What happens when a 1 cannot be determined on a row?

5.3. Start at the bottom of the final ACM when interpreting the final solution.

5.4. How the General solution is to be written with parameters and also a couple of particular solutions

5.5. how to avoid fractions until the last ERO is to be done. You do not have to get the ones first. Get the zeros first if it helps to avoid fractions (see 5c in notes)

6. First Find The Augmented Matrix of the following examples: