# Reduced Row Echelon Form – #1 Skill in Linear Algebra

Reduced Row Echelon Form or RREF is the skill you MUST know for Linear Algebra.

Why is it so important?

Because every topic covered in Linear Algebra requires you to row reduce a matrix.

Row Reduction allows us to solve systems of equations quickly and easily and enables us to determine linear independence, orthogonality, vector spaces, and so much more.

Trust me, just by knowing how to row reduce will give you the Edge you need for everything you will encounter in Matrix Algebra.

But don’t worry, you have been row reducing since Algebra 1! Remember when you were asked to solve a system of equations, and you had to either use substitution or the elimination method (i.e., linear combinations)?

Well, guess what! Every time you performed the elimination method and solved the system, you row reduced!! We wrote our steps differently, but the process is exactly the same! Phew!

So what’s the difference between what we did for solving systems in Algebra 1 and Linear Algebra?

Well, it’s all about simplifying the process and allowing us to solve bigger systems with more equations and variables, quickly and easily!

Here’s the RREF Overview that I explain in the video:

I begin with an understanding of the similarities and differences between Row Echelon Form and Reduced Row Echelon Form for matrices.

Row Echelon Form (REF) Objective – find out what we want to accomplish, and our overall goal is when we are performing Row Echelon Form or Gauss Elimination

4:15 Reduced Row Echelon Form (RREF) Objective – learn what our end result will be when we perform Reduced Row Echelon Form or Gauss-Jordan Elimination

The idea behind these two forms is that you want to take a system of equations and transform it, through a series of steps so that you can solve the system (i.e., solve for each variable). Sometimes all we need is to determine how many pivots (basic) variables we have, and other times we need to solve completely. Therefore, we have two types of forms: REF and RREF.

Wait! What’s a Pivot or Basic variable?

8:39 Pivot (Basic) variables are dependent variables, and can be easily found via row reduction. In fact, with row reduction, we will also be able to determine which variables are independent (free).

So how do we row reduce?

14:32 Row Operations – there’s only three!!

1) Replace – you can ADD two rows together.
2) Interchange – you can switch two rows
3) Scale – you can multiply an entire row by a number (scalar)

Here’s a big, big hint…. you can only add, because addition is commutative. So if you need to subtract two numbers, then we just add a negative!

18:19 Example 1 – Here I walk you through solving a system of equations first by using the Elimination (Linear Combination) Method from Algebra 1, and then I show you how we do the exact same thing using Matrix Row Operations.

40:37 Example 2 – We perform the necessary steps for Row Echelon Form to solve the system.