Determinant

Defined only for square matrices. Denoted by .

For 2x2

For higher order

Minor of an element

Suppose .

Minor of an element , is the matrix obtained by deleting row and column of . Denoted by .

Co-factor of an element

Suppose .

Co-factor of an element , is defined as (commonly denoted as ):

Definition

If then the determinant of is defined by:

For some .

Properties of determinants

• Every element of a row or column of a matrix is then the value of its determinant is .
• If 2 columns or 2 rows of a matrix are identical then its determinant is .
• If A and B are two square matrices then .
• The value of the determinant of a matrix remains unchanged if a scalar multiple of a row or column is added to any other row or column.
• If a matrix is obtained from a square matrix by an interchange of two columns or rows: .
• If every entry in any row or column is multiplied by , then the whole determinant is multiplied by .

Composition

In relation with eigenvalues

For a matrix A with number of eigenvalues: