Determinants

Definition 3.1: Each - square matrix is assigned a special scalar called the determinate of, denoted by or, or

.

Determinant of order 1: ;

Determinant of order 2: ;

Determinant of order 3:

Definition 3.2: Minor of an element of determinant is a determinantobtained from determinant deleting its -th row and -th column and denoted.

Definition 3.3: A quantity is called cofactor of an element of determinant and denoted

.

Theorem (Laplace expansions): The determinant of a square matrix is equal to the sum of the products obtained by multiplying the elements of any row (column) by their respective cofactors

§4. Solution of system of linear equations in unknowns.

The system is a system of linear equations in unknowns. – is a matrix of the coefficients of unknowns of system, – is a determinant of this matrix, – matrix -column of constant terms.

1. Cramer’s method.

Theorem 4.1: If , the system has unique solution, which is defined by the formulas:

, , …, ,

where determinants are received from by replacing the -th column of by the column of constant terms.

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