How Do Pivot Columns Influence the Existence of Free Variables in a Matrix?

[karush]

ok a pivot column is one of $[0,,,0,b]$ where b is non zero... b is 1 in RREF
not sure of the best answer but #23 there will be no free variables since 4 equation can derive fout answers altho some asnwera man be the same
#24 we can not derive a 4th or 5th answer with 3 equations

 

[jvicens]

Hello there,

I would like to clarify a few things about pivot columns and free variables in the context of a matrix.

Firstly, a pivot column is a column in a matrix that contains a pivot element, which is the first non-zero element in that column. In the case of a pivot column being $[0, 0, ..., 0, b]$, the pivot element is the non-zero value of b. This column is important because it helps us determine the leading variables in a matrix, which are the variables that are not dependent on any other variables.

In a reduced row echelon form (RREF) matrix, the pivot elements are always 1, as you correctly mentioned. This is because in the process of reducing a matrix to RREF, we use row operations to make the pivot element 1 and eliminate all other elements in the same column.

Now, onto your question about free variables. In a matrix, a free variable is a variable that can take on any value, as it is not dependent on any other variables. In the context of a system of equations, a free variable would correspond to a parameter in the solution set.

In the case of a 4x4 matrix, if there are 4 pivot columns, then there are no free variables. This is because each pivot column corresponds to a leading variable, and all other variables can be expressed in terms of these leading variables. Therefore, there can only be a unique solution to the system of equations represented by this matrix.

If there are less than 4 pivot columns, then there will be free variables. This means that the system of equations represented by the matrix will have infinitely many solutions, as the free variables can take on any value.

I hope this helps clarify any confusion. Let me know if you have any further questions.