Determinant of the square matrix

For every square matrix A of order nwith real or complex elements we can uniquely associate a real or complex number D, which is called the determinant of А. The general expression for the determinant matrix n-th order is usually given in the form:

det[A] =

a11 a12 . . . a1n a21 a22 . . . a2n . . . . . . . . . . . . an1 an2 . . . ann

= Σ(-1)e a1α1a2α2. . . anαn     (1)

At the right side there is a sum of compositions in form like a_1α1a_2α2. . . a_nαn Each of these compositions must contain elements of the matrix a_ij aijwhich are located in different lines and different columns. In other words, it must contain one element from every line and every column. This means that all of the first just like all of the second indexes must differ.
If we'll place first indexes in the ascending order, like it have been done above, then the aggregate of second indexes will form permutation (α₁, α₂, ..., α_n) of set of numbers from 1 to n. Because the number of all permutations from n numbers equals n! (n factorial), the it is possible to form the same quantity of compositions a_1α1a_2α2. . . a_nαn from the present matrix elements (with zero elements, some of themare zero). Determinant is equal to the sum of all such compositions made with the sign (-1)^e where е - - the number of inversions in the permutation of second indexes (α₁, α₂, ..., α_n). Instead of the factor (-1)^e can write the sign sgn(α), which is positive for an even number of inversions and negative for odd number of inversions in the permutation of second indexes (α₁, α₂, ..., α_n).
The order of the determinant is the same order as its matrix. The elements a_ij of А matrix is also called "the elements of the determinant |А|". Compositions (-1)^ea_1α1a_2α2. . . a_nαn are called "the members of the determinant".

It is easy to transform general determinant calculation formulas into determinant of the any order calculation formulas. So, for the determinant of the second order we'll get next formula:

det[A] =

a11 a12 a21 a22

= a11a22 - a12a22     (2)

Similarly, for the determinant of the third order:

det[A] =

a11 a12 a13 a21 a22 a23 a31 a32 a32

= a11a22a33 +a12a23a31 +a13a21a32 -a13a22a31 -a11a23a32 -a12a21a33      (3)

As it can be seen, Indexes of the columns of all members of the determinant of the third order are determined by permutations (1, 2, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), (1, 3, 2), (2, 1, 3), number of inversions are equal to 0, 2, 2, 3, 1, 1. The general expression of the determinant of n-th order is useful for study and for proof of its properties but for the manual computing of determinants there're other more practical methods based on the properties of determinants.

Minors and cofactors

If D = |A| - determinant of the order n, then the minor M_ijof the element а_ij is determinant of the order n-1, which was obtained by crossing out i-th line and j-th column out of D. Cofactor A_ij of the element а_ij is the minor M_ij, multiplied by (-1)^i+j, i.e. A_ij = (-1)^i+jM_ij

E.g. for the determinant of the third order:

D =

a11 a12 a13 a21 a22 a23 a31 a32 a32

Minor of the element а23 will be M23 =

a11 a12 a31 a32

Cofactor of the element а₂₃ will be A₂₃= (-1)²⁺³M₂₃= (-1)⁵M₂₃= -M₂₃
And for the element а₁₃ , A₁₃= M₁₃ и т.п.

The decomposition theorem

If D = |A| - determinant of the n-th order, then

D =

n Σ i=1

aikAik

=

n Σ i=1

akiAki

n Σ i=1

aikAim

=

n Σ i=1

akiAmi = 0,  для k неравного m

i.e. sum of the compositions of all the elements of a line (or column) and the relevant cofactors is equal to the value of the determinant. Sum of the compositions of all the elements of a line (or column) and the cofactors of therelevant elements of another line (or another column) is equal to zero.

Main determinant properties

First of all, det[A] = det[A]^t, i.e. matrix determinant remains the same in spite of mutual replacement of lines and columns. So, all of the determinant properties that are true for its columns are also true for its lines, and vice versa.

The following are the basic properties of determinants, which can easily be proved on the basis of the general expression (1).

1. When two columns of the determinant are switched it changes its sign to the opposite one (anti-symmetry property).

2. Determinant equals zero if all of the elements of any column are equal tozero, or if one of the columns is a linear combination of any of its other columns (in particular, the determinant which hasat least two same columns is equal to zero.).

3. Multiplication of all of the elements of any column by a scalar k is equivalent to the multiplication of the determinant by k (a common multiplier of elements of theline or column can be put beyondthe sign of the determinant).

4. Matrix of nth order multiplication by a scalar K matches the multiplication of the determinant by the Kⁿ, i.e.
det(k[A]) = kⁿdet[A].

5. Value of the determinant will not change if we add a column to any other column multiplied by a scalar K.

6. If two determinants of the same order differ only in the elements of j-th column, then its sum is equal to the determinant whose elements of the j-th column is equal to the sums of corresponding elements of the j-th column of original determinants, and other elements are the same as in the original (the linearity property).

Determinant calculation

The determinant of order 2 can be easily calculated by using the formula (2). To find the value of the determinant of the third order, you can use the formula (3). Determinants of higher orders could also be calculated like that,however, this requires a lot of effort. Often it's done so: the determinant of n--th order is transformed into a determinant of the (n-1)-гth order, then it's transformed into a determinant of (n-2)-th order, and so on, until, finally, we will not receive the determinants of 3rd or 2nd order. The basis of this "gradual reduction of order" principle is the decomposition theorem: the determinant of n-th order D is written as a sum of determinants of order n-1 ("is decomposedby the elements of i-thlibe and j-th column"). To each of these determinants of order n-1 can be applied to the decomposition theorem.

If all the elements а_ik i--th line of the determinant D, except for one, are equal to zero, the amount received after the application of the composition theorem, contains only onenon-zero summand. If before the decomposition of the determinant by the elements of i-th line, those elements are turned into zeros, the calculation will be simplified. This is possible thanks to the properties of the determinants (especially property 5).

Even more convenient thing is the computation of the determinant, if you can convert it so that all the elements at the left and below the diagonal а₁₁ , а₂₂, ..., а_nn are equal to zero. As can it can be easily to understood from the decomposition theorem, the determinant is obtained as the composition of the terms on the main diagonal:   D = a₁₁a₂₂.. .а_nn .