Linear Algebra( Row-Reduced Echelon Form)

So what is Row reduced Echelon form ? And where this used ? two questions . Below will be simple explanation for the same . Hope this will be helpful .

RREF is used to solve system of linear equations. The manipulations that we might do to an augmented matrix to try to solve it. But question “when do you stop doing the row operations?” the answer is “RREF”, there are some rules which defines whether a matrix is in RREF or not to find solution .

Here are those 4 rules :-

1. The first rule is that any row of all zeros has to be at the bottom only .
2. The second rule is that if you have a non-zero row, then the first entry in that row has to be a one. These Ones are called “Pivots”
3. The third rule is that for any pivot, it has to be the only non-zero entry in its column.
4. The fourth rule is that any pivot in a lower row has to be to the right of any pivot in a higher row, or in other words, the pivots just go down and to the right.

So those are the four rules for whether a matrix is in RREF. They’re kind of weird, but you’ll get used to them.

“So what kind of manipulations we can perform on matrix to form RREF” ?

There are three types of elementary row operations that one can apply to a matrix.

1. Switch two rows.
2. Multiply a row by a non-zero scalar.
3. Add a scalar multiple of one row to another row. (More precisely, replace one row with the sum of itself and a scalar multiple of a different row.)

May be you have question that why we always multiply row by non-zero number only ?

Ans :- if we multiply by zero scalar then we can’t make back propagation to get original linear equation from RREF . You can try and think for the same with some examples :)

Examples :- A is RREF but not B . Why not B ? Check out above points related to RREF and find the answer :)

I hope you enjoyed my article. Thank you for reading!

References

References and useful links:

Row Echelon Form & Reduced Row Echelon Form - Statistics How To

Reduced row echelon form