# Determinant

A determinant is a function in linear algebra that associates a number with each square matrix. The determinant is defined only for square matrices.

The determinant of a second-order square matrix can be found using the formula:

\begin{align} \textrm{det}(A)=\begin{vmatrix} a & b\\ c & d \end{vmatrix}=ad-bc. \end{align}

The determinant of a third-order square matrix can be found using the formula:

\begin{align} \textrm{det}(A)&=\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}=\\ \\ &=a_{11}a_{22}a_{33}+ a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} − a_{13}a_{22}a_{31} − a_{11}a_{23}a_{32} − a_{12}a_{21}a_{33} \end{align}

Generally, the determinant of an n×n matrix can be calculated using the Leibniz formula or the Laplace's expansion formula.

## Algebraic complement, subdeterminant, and minor

The algebraic complement of an element aij, denoted as Aij, is called the minor of that element. It is taken with a sign of "+" when the sum of indices i+j is an even number, and with a sign of "-" when it is an odd number.

The minor of an element aij in matrix A, denoted as Mij, is the determinant of the matrix obtained by removing the i-th row and j-th column from matrix A.

Let's consider a 3x3 matrix:

\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & {\color{Red} a_{{\color{Red}2}{\color{Red}3}}} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}

The minor M2,3 of this determinant can be found using the subdeterminant:

\begin{pmatrix} a_{11} & a_{12} & * \\ * & * & {\color{Red} *} \\ a_{31} & a_{32} & * \end{pmatrix}

$$M_{2,3}= \textrm{det} \begin{vmatrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{vmatrix} = a_{11}a_{32}-a_{12}a_{31}$$

## Leibniz Formula for Determinant Calculation

Take the sum over all permutations σ from the set {1, 2, ..., n}:

$$\textrm{det}(A)=\sum_{\sigma \in S_{n}} \textrm{sgn}(\sigma )\prod_{i=1}^{n}A_{i,\sigma_{i}}$$

## Laplace's Formula for Determinant Calculation

According to Laplace's formula, the value of the determinant is equal to the sum of products of the elements of any row and their corresponding algebraic complements.

$$\textrm{det}(A)=a_{i1}A_{i1}+a_{i2}A_{i2}+...+a_{in}A_{in}$$

where,

a— matrix element;
A— algebraic complement of the corresponding element.

Expressing the algebraic complement through the minor, we will have:

$$\textrm{det}(A)=\sum_{i=1}^{n}(-1)^{i+j}a_{ij}M_{ij}$$

## Properties of Determinants

1. The value of a matrix determinant does not change when the matrix is transposed:

$$\textrm{det}(A)=\textrm{det}(A^{T})$$

2. The determinant is zero if:

• it consists of zeros,
• it is equal to another corresponding row or column,
• it is proportional to another corresponding row or column,
• it can be expressed as a precise sum of scalar multiples of the remaining rows/columns.

3. If two rows are exchanged in the determinant, the sign of the determinant becomes opposite:

\begin{align} \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}&=n\\ \\ \begin{vmatrix} a_{21} & a_{22} \\ a_{11} & a_{12} \end{vmatrix}&=-n\\ \end{align}

4. When multiplying the determinant by a scalar, only one row or column is multiplied. Similarly, the reverse holds true: if a row or column of the determinant is expressed as a scalar multiple, that scalar can be brought in front of the determinant.

5. If each element in a row (column) of the determinant is the sum of two addends, the determinant can be expressed as the sum of two determinants. In the first determinant, consider the first addends in the corresponding row (column), and in the second determinant, consider the second addends in the corresponding row (column), with the remaining rows (columns) being the same as in the original determinant.

6. The value of the determinant remains unchanged when any multiple of the corresponding elements of one row is added to the corresponding elements of another row.

7. Since the determinant is defined inductively (starting with the first order, then the second, and so on), larger determinants can be calculated as the sum of their minors or subdeterminants.

8. The product of the determinants of matrices A and B is equal to the determinant of their matrix product, regardless of the order of the matrices:

$$\textrm{det}(A) \times \textrm{det}(B)=\textrm{det}(AB)=\textrm{det}(BA)$$