Matrices : reduced row echelon form

In this page, i'm going to talk about matrices.

 

1. If the number of equations are less than the total number of variables(unknowns), then you will have an infinite number of solutions for the equations.

 

2. The idea of matrices : The matrices are arrays of numbers. 

 2-1) the coefficient matrix : It literally uses coefficients of equations when you describe a matrix. As we use matrices, we can reduce unnecessary time to write variables like x1, x2 etc,. It means this way gives us efficiency.

 2-2) the augmented matrix : when it says augmented matrix, it means that you put coefficients of variables into a matrix and draw a vertical line to distinguish the area between the coefficients and the values for equations. It's how it looks like.

 2-3) reduced row echelon form : leading coefficient in any of rows is 1 and every other entry in that column is 0. This form is referred to as reduced row echelon form. This is how it looks like.

 2-4) pivot entry : the only non-zero entry in a column.

** In this situation, entry means only one number, not a row, not a column. For example, the first entry in the first row is 1, not (1, 2, 0, 3).

** You can see the zeroed out line in the matrix. When you do a reduced row echelon form, the zeroed out line has to be last row in your matrix. It's the convention.

** the leading entry in each successive row is to the right of the leading entry of the previous row.

 2-5) pivot variable : it means variables associated with pivot entries.

 2-6) free variable : it's variables not associated with pivot entires.

 ** you can set to any value when you solve free variables. So, you can only solve for pivot variables.

 

3. No solutions : when you put a matrix in reduced row echcelon form, you're going to get a statement that zero is equal to something, and this means that there is not solution.

** If you have the situation where you have the same number of pivot entries as columns, then you have a unique solution.