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 ith row and jth 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.

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