1 Project Statement Problem Use Theorem 1210 to show that re

1 Project Statement Problem: Use Theorem 1.2.10 to show that reduced row echelon forms are unique. Then, prove Theorem 1.4.1 and 1.4.2 in details

Solution

With the help of the definition of row echelon form the theorem 1.4.1 is clear

When we change the matrix \' A \' to row echelon form

step 1 . first we will make a₁₁ is a non zero element  ( element in first row first column , preferably 1 )

and the operation are made for the other rows

so that all the other element in the 1 st column below a₁₁ are o ( which can be done in a unique way )

step 2. same way we see to that a₂₂ is non zero and the row operations are done so that

the elements below a₂₂ are zero and so on which also can be done in a unique way   

when we adapt the method to change to row echelon form we get a unique matrix equivalent to A