# Definition of Matrices

### Introduction

Matrices (or matrix in singular), is a compact notation method for element arrays. Often matrices are used with equations systems and vectors. The elements can then be easily treated.

Consider the following system of algebraic equations:

# Characteristic Equation, Eigenvalues

The characteristic equation of a square matrix A is:

where λ is a scaling factor.

This returns a polynomial, which is called the characteristic polynomial:

# Derived Forms of Matrices

### Matrix Transpose

In a matrix, if we change the position of the elements so that the rows become columns and the columns become rows, we get the transpose of the initial matrix.

# Determinants

### Defintion

Determinant is a number that is derived from a square matrix using an algorithm. If the determinant is other than zero, it means that the associated equations system has a unique solution or the associated vectors are linearly independent . The determinant of a square matrix is denoted as:

# Operations of Matrices

In order to add two matrices A and B, they must have the same number of rows and columns. Each element of the first matrix aij is added with the corresponding element of the second matrix bij and the result is an element cij in the same position in the resulting matrix.

# Solving Equation Systems

### Introduction

There are mainly two methods of solving equation systems using matrices. The first uses the inverse matrix of the coefficients of the independent variables and the second converts the coefficients matrix into a unit matrix form.