The determinant is a real function such that each square matrix joins a real number (the determinant of a matrix ). The determinant of a square matrix is denoted as or .
The determinant of matrices we define as inductive, that is, the determinant of a square matrix of the -th order we define using the determinant of a square matrix of the -th order.
The determinant of a matrix of order is the number :
The determinant of a matrix of order 2, , is the number:
The determinant of a matrix of order 3,
, is the number:
The determinant of a matrix
of the -th order is the number
Calculate the determinant of the following matrices:
As we can see, calculating the determinant it is not easy for matrices of a higher order. There are several methods for calculating the determinant. The general method is the Laplace expansion of a determinant along a given column or row.
Let be a square matrix of order . If in a matrix is removed -th row and -th column, we have thus obtained a matrix of order of which the determinant is called a minor and is denoted as .
The algebraic complement or cofactor of an element is the number:
Let be an matrix. For a matrix we have the following expansions:
a) for a fixed :
b) for a fixed :
The properties of determinants
Let be a matrix of order .
1.) A matrix and its transpose matrix have the same determinants, that is
2.) If is a triangular matrix, then its determinant is equal to the product of all diagonal elements, that is
3.) If two columns or rows of the determinant interchange the location, then
4.) If a matrix has two equal columns or rows, then .
5.) If is a matrix with a row or column where every element is equal to zero, then .
6.) The determinant value does not change if to one column we add a linear combination of the remaining columns.
7.) If a matrix is obtained from a matrix by multiplying one its row (or column) with a scalar , then
8.) A matrix is a regular matrix iff .
9.) For a singular matrix is .
Using the Laplace expansion of a determinant along -th column, calculate the determinant of the following matrix:
In general, we choose the row or column that contains the most zeroes, because then we have a shorter calculation.
The Binet – Cauchy theorem.
If matrices and are square matrices of the same order, then
We can observe that for a regular matrix
is valid if we know the determinant of the matrix and through using the Binet – Cauchy theorem we can calculate . Namely,
The adjugate or adjoint of a square matrix is a matrix , where , . This means that the adjugate of a square matrix is the transpose of a cofactor matrix of a matrix .
The adjugate of a matrix can also be denoted as .
For instance, the adjugate matrix of a matrix is a matrix .
As was mentioned previously, a square matrix is a regular matrix iff . In this case, an inverse matrix of a matrix is given by the following formula:
Now we can find, using the formula above, an inverse matrix of a given square matrix of any order.