Matrices intro

Matrices are just a rectangle array of numbers.
Prerequisites: Systems of equations

Matrices could be seen as a group of informations arranged IN A CERTAIN WAY.
IT’S SO SO SO SO SO EASIER TO THINK IT AS A SYSTEM OF EQUATIONS.

Matrix row operations & Systems of equations

It’s very SAME with operations of systems of equations.

Refer to Khan academy article: Matrix row operations

There’re different types of row operations:

• Switch any two rows
• Multiply a row by a nonzero constant
• Add one row to another

They all relate to the operations of systems of equations:

Switching any two rows:

Multiply a row by a nonzero constant:

Add one row to another:

Solve system equations using Matrix

it’s also called the Row-Echelon form and Gaussian elimination.

Khan lecture: Reduced row echelon form
Refer to Ck-12: Row Operations and Row Echelon Forms
Example of “RREF”: Lec 01 — Linear Algebra | Princeton University

It’s a so serious problem in all the first lesson of Linear algebra courses. It seems simple yet not easy to solve by yourself. You need to understand all the steps of how to do a REF or RREF, aka. Reduce Row Echelon Form.

The important note to apply the RREF is to know how the Pivot, or the Cursormoves.
It's more efficient to understand it with 1 or 2 practice rather than see notes here.

First we need to rewrite the system of equation to matrix form:

Then by row operations, we need to achieve this kind of result, which is also called Reduced Row Echelon Form:

It means we eliminated all other variables and only left 1 variable in one equation, which is called Identity Matrix. Then you could put back the number to the system of equations.