Systems of Linear Equations

A linear system of equation is consistent when it has at least 1 solution. Otherwise, it is inconsistent.

For the matrix

The coefficient matrix, often denoted by :

The constant matrix (or constant vector), often denoted by

The augmented matrix, denoted by

We can:

• Swap two rows
• Add a scalar multiple of one row to another
• Multiply any row by a non-zero scalar

The first non-zero entry of each row is called a leading entry or pivot

A matrix is in row echelon form (REF) if :

1. All rows whose entries are all zero appear below all rows that contain nonzero entries
2. Each leading entry is to the right of the leading entries above it

The process of reducing to REF is known as Gaussian Elimination

A matrix is in reduced row echelon form (RREF) if it is in REF and:

1. Each leading entry is a 1 (leading 1)
2. Each leading one is the only nonzero entry in its column

The process of reducing to RREF is known as Gauss-Jordan Elimination

A free variable is just a parameter from a solution

Since columns 1 and 3 do not have a leading 1: