Definition of Determinants
Determinants are functions
with quadratic matrix variables and real values. In this article we begin the
study of real value functions of a matrix variable, which is a function that
associates a real number f (x) with
an X matrix. Assessment of a function
like that is called a determinant function.
The definition of
determinant will be given inductively using minor and cofactor definitions.
Utilizing the deteminan properties associated with elementary row operations,
an easy way to calculate determinants is formed. On this occasion we will also
discuss the application of determinants in solving a linear system using the
Cramer rules.
Each square A matrix is
always associated with a scalar called deteminan the matrix, and we write it
as det (A) or A. Before starting with a more general one, we first take a
matrix of size (2 x s) as follows:
Defined:
Example 1
Note: We must distinguish
parentheses for the matrix "[]" with the determinant "  "
Before explaining
definitive determinants, it is first discussed about permutations, inversions and
their classifications, and the results of elements.
Definition of Permutation,
A permutation of a set of integers {1, 2, ..., n} is an arrangement of these integers according to a rule without omitting or repeating these numbers.
Example 2.
There are six different permutations of the set of integers {1, 2, 3}. These permutations are:
(1, 2, 3) (2, 1, 3) (3, 1, 2)
(1, 3, 2) (2, 3, 1) (3, 2, 1)
One method that is
systematically and easily used to compile the results of permutations is to use
a permutation tree.
Generally, the set (1, 2,
..., n) will have n (n1) (n2) ... 2.1) = different permutations. To declare a
general permutation from the set {1, 2, .., n}, we will write (j_{1}, j_{2}, ..., j_{n}).
Here, j_{1 }is the first
integer in permutation, j_{2}
is the second integer in the permutation and so on. An inverse is said to occur
in a permutation (j_{1}, j_{2}, ..., j_{n}) when a larger integer precedes a smaller integer.
Example 3.
Permutation (4 3
2 1)
(j_{1},
j_{2}, j_{3} , j_{4)}
There are five inversions, namely:
 j_{1} = 4 precedes j_{2} = 3, even though 3 <4
 j_{1} = 4 precedes j_{3} = 1, whereas 1 <4
 j_{1} = 4 precedes j_{4} = 2, even though 2 <4
 j_{2} = 3 precedes j_{3} = 1, even though 1 <3
 j_{2} = 3 precedes j_{4} = 2, even though 2 <3
Definition A permutation is called even if the total number of inverses is an even integer and is called odd if the inverse is an odd whole number.
Example 4.
The table below will classify various permutations
{1, 2, 3} as even or odd.
Permutation

Number of Inversions

Classification

(1, 2, 3)

0

Even

(1, 3, 2)

1

Odd

(2, 1, 3)

1

Odd

(2, 3, 1)

2

Even

(3, 1, 2)

2

Even

(3, 2, 1)

3

Odd

Review the matrix n x n
What we mean by the elementary multiplication result of A is every product of n entries from A, which cannot have two factors coming from the same line or from the same column.
Example 5.
Arrange all elementary multiplication results from matrices.
(i) because the factor of every elementary multiplication has
two factors, and because factors come from different rows, an elementary
product can be written in the form of:
a_{1}_
a_{2}_
where blank points indicate column numbers. Since there
are no two factors in the multiplication result coming from the same column,
then the column must be 1 2 or 2 1. Then the
elementary multiplication results are only a_{11}a_{22} and a_{12}a_{21}.
(i) Because each elementary
product has three factors, each of which comes from a different line, an
elementary product can be written in form.
a_{1}_
a_{2}_a_{3}_
because there are no two
factors in the multiplication coming from the same column, then the column
number has no repetition, as a consequence: the column numbers must form a
permutation of the set {1, 2, 3}. Permutation 3! = 6 this produces the
arrangement of elementary multiplication results below,
a_{11}a_{22}a_{33 } a_{12}a_{21}a_{33} a_{13}a_{21}a_{32}
a_{11}a_{23}a_{32} a_{12}a_{23}a_{31} a_{13}a_{22}a_{31}
As shown by the example
above, an A matrix of size n x n has n! Elemental multiplication results. The
elementary multiplication results are multiplications in the form of a_{1j1}a_{2j2}...a_{njn},
where (j_{1}, j_{2}, ..., j_{n}) is a permutation of the set (1, 2, ..., n). What we
mean by an elementary product multiplied by A is an elementary product of a_{1j1}a_{2j2}...a_{njn}
multiplied by +1 or 1. We use the "+ (plus)" if (j_{1}, j_{2}, ..., j_{n})
is an even permutation and the " (negative)" if (j_{1}, j_{2}, ..., j_{n})
is an odd permutation.
Example 6.
Arrange all elementary multiplication results marked from matrices into a table.
(i)
Elemental
Multiplication Results

Permutation
associated

Even or
Odd

Elemental
Multiplication marked

a_{11}a_{22}

(1, 2)

Even

a_{11}a_{22}

a_{12}a_{21}

(2, 1)

Odd

 a_{12}a_{21}

(ii)
Elemental
Multiplication Results

Permutation
associated

Even or
Odd

Elemental
Multiplication marked

a_{11}a_{22}a_{33}

(1, 2)

Even

a_{11}a_{22}a_{33}

a_{11}a_{23}a_{32}

(2, 1)

Odd

a_{11}a_{23}a_{32}

a_{12}a_{21}a_{33}

(1, 2)

Even

a_{12}a_{21}a_{33}

a_{12}a_{23}a_{31}

(2, 1)

Odd

a_{12}a_{23}a_{31}

a_{13}a_{21}a_{32}

(1, 2)

Even

a_{13}a_{21}a_{32}

a_{13}a_{22}a_{31}

(2, 1)

Odd

a_{13}a_{22}a_{31}

Let A be a quadratic
matrix. The determinant of matrix A, expressed by det (A), is defined as the
sum of all elementary multiplication results marked with A.
Example 7.
From the example above, to be able to more easily remember the formula of the results of elementary multiplication, then consider the one below,
Determinants are then calculated by summing the multiplication results in the arrows that point to the right and subtracting the arrows pointing to the left. So, we conclude that the determinant of A is often symbolically written as
Where ∑ indicates that the terms must be added to all
permutations (j_{1}, j_{2}, ..., j_{n}) and the symbols "+" or "" can be
selected in each tribe according to whether the permutation is even or odd.
Example 8.
Calculate the determinants of
Using the method above will get a value,
det(A)=3 x (2)
–(1)x4=10
det(B)=(1X5X9)+(2X6X7)+(3X(4)X(8))(1X6X(8)(2X(4)X9)(3X5X7)
det(B)=45+84+96105(48)(72)=240
The method as above, specifically for a 3 x 3 matrix is called the SARRUS Method
Note:
The method above does not apply to determinants of 4 x 4 matrices or for higher matrices. Direct calculation of a 4 x 4 determinant will involve a calculation of 4! = 24 elementary multiplication results marked. Thus, the remaining part will be developed using determinant properties that will simplify the calculation.
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