Determinants

Abigail Huettig

28 Determinants

Determinants are a special characteristic of square ( $n\times n$ ) 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 $A$ is denoted as $|A|$ or det( $A$ ), and the determinant of a $2\times 2$ matrix

$\begin{bmatrix} a&b\\ c&d \end{bmatrix}$

is calculated as $ad-bc$ [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 $A$ is a square matrix, then the minor $M_{ij}$ of the element $a_{ij}$ is the determinant of the matrix obtained by deleting the $i$ th row and $j$ th column of $A$ . The cofactor $C_{ij}$ is given by

$C_{ij}=(-1)^{i+j}M_{ij}.$

Theorem VI.3 (Expansion of Cofactors)

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

$det(A)=|A|=\sum_{j=1}^n a_{ij}C_{ij}=a_{i1}C_{i1}+a_{i2}C_{i2}+...+a_{in}C_{in}$

or

$det(A)=|A|=\sum_{i=1}^n a_{ij}C_{ij}=a_{1j}C_{1j}+a_{2j}C_{2j}+...+a_{nj}C_{nj}.$

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.

28 Determinants

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