# 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:  The calculation of a determinant of a 2X2 matrix is: The determinant of 3X3 matrix is calculated by breaking it down to a summation of 2X2 determinants as follows:   The 3X3 determinant can be also calculated across any row or column instead of the first row. For example, we could calculate it across the third column:  The sign is plus or minus for each element of the summation, according to the formula: So the element a is positive because i=1 and j=1. The element b is negative because i=1 and j=2.

For larger determinants, using the same method we break them down to summation of smaller and smaller determinants, until they become at size 2X2 and then they can be calculated.

### Example

For a numerical example we will use the following matrix:           