Matrices

Matrices are a 2D arrangement of numbers which can be used to represents systems of equations through an augmented matrix.

This is represented as the form $\left[\begin{array}{c}A|b\end{array}\right]$. Where $A$ is the coefficient matrix and $b$ is the vector of solutions.

If $\vec{b}$ is the $\vec{0}$, then the system is called a Homogeneous System.

Elementary Row Operations

Matrices have the ERO which can be used to eliminate rows and solve systems of equations.

1. Multiply a row by a non-zero scalar
2. Swap rows
3. Add a multiple of a row to another

REF and RREF

The point of doing the elementary row operations is to take a matrix to one of these forms. These are Row Echelon Form and Reduced Row Echelon Form.
REF

1. All zero rows must appear under non-zero rows
2. The rows must be in order of leading coefficients

RREF

1. All leading entries must be 1’s
2. In a column with a leading 1, all other entries are 0

Solution Sets

A system of equations has 3 possibilities.

1. No solution.
2. Infinitely many solutions, here the entire row is 0.
3. Uniquely many solutions, here the row is not all 0’s and the augmented side is non-zero. I.e, if all of the variables have a leading entry in REF.

Rank

If the matrix is in RREF, the number of leading 1s is called the rank of the matrix.

Related

• Matrix Operations
• Matrix Mappings
• Diagnonalization