How to compute row-reduced echelon form and understand upper triangular matrices

[innightmare]

I am having problems with understanding the whole concept/how to compute the row-reduced echelon form.

Can someone please help me? Thanks

 

[bel]

A matrix remains unchanged after going through the elementary row operations, so the whole concept is to keep on multiplying rows and adding (or subtracting) them from other rows to give an upper triangular matrix.

 

[innightmare]

The book that i have doesn't give examples nor is it clear about the upper triangular matrix. Can you PLEASE explain what's an upper triangular matrix?

 

[bel]

It is just a matrix \{a_{ij}\} where the terms for which i is bigger than j are all zero.

 

[innightmare]

yes, but i thought you changed your matix after changing the equation pertaining to it

 

[bel]

Yes it does, generally, but not if you change the system of equations in strict accordance with the elementary row operations. Chapter three of Wylie's and Barrett's Advanced Engineering Mathematics (sixth edition) has proofs, and most university libraries have that book, I think.

 

[ice109]

just write out a system of equations, any system which you know is consistent and solve it. now write out the matrix for it and get it into rrref form and you'll see that you're performing the same operation you've just taken out the xs

 

[HallsofIvy]

The book that i have doesn't give examples nor is it clear about the upper triangular matrix. Can you PLEASE explain what's an upper triangular matrix?

An upper triangular matrix is a matrix that has only zeros below the "main diagonal".
In other words, the non-zero entries form a triangle and it is above the diagonal.