Determinants, their properties. Minors and cofactors. Lecture 2

A(n; n) is the determinant obtained from the given one by deletion of the i-th row and the j-th column.
For a determinant of the second order
Example: Δ=
For a determinant of the third order
and etc.

cofactor А of the element а of a square matrix А(n; n) is the minor multiplied on the number , i.e.
Properties of determinants
Property 1. The value of a determinant doesn’t change if we replace all its rows by the corresponding columns and vice versa.

the determinant will change a sign.
Property 3. If all the elements of a row (column) of a determinant are multiplied on the same non-zero number «m» then the determinant value increases (decreases) in «m» times.
Example:
Corollary 1. If all the elements of a row (column) of a determinant have a non-zero common multiplier, it can be taken out the determinant sign.

(columns) are proportional respectively is equal to zero.
Property 4. Let each element of an arbitrary row (column) be the sum of two addends. Then the determinant is equal to the sum of two determinants such that the corresponding row (column) of the first determinant consists of the first addends, and the corresponding row (column) of the second determinant consists of the second addends.
Example: For example, let the first column of the determinant be the sum of two addends. Then