# 28 Determinants

Determinants are a special characteristic of square () matrices. Every square matrix has one, and there are different ways to calculate it based on the size of the matrix and pattern of the elements within the matrix [16]. Sometimes, determinants tell us about the matrix itself, like if it is invertible or not [16]. Other times, they are used to solve systems of equations [16]. They have many applications to geometry, such as finding area, volume, and equations of lines and planes [16]. They are also useful in finding eigenvalues and eigenvectors which have many interesting real world applications themselves [16]. There is so much we could discuss related to this topic, but we will only be able to cover a few key ideas.

The determinant of a matrix is denoted as or det(), and the determinant of a matrix

is calculated as [16]. However, as the matrix gets bigger, finding the determinant becomes more complicated. Consider the following definition and theorem from Larson and Falvo [16].

Definition VI.2
If is a square matrix, then the minor of the element is the determinant of the matrix obtained by deleting the th row and th column of . The cofactor is given by

Theorem VI.3 (Expansion of Cofactors)

Let be a square matrix of order . Then the determinant of is given by

or

See example 63 in chapter 30 to see how this definition and theorem can be used to find the determinant of a square matrix.

While matrices are necessary for linear algebra, they are not all there is. They are actually just a subset of the much larger set of vectors.