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:

 

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