Determinants, Cofactors of aij in |A|, Properties of Determinants, Evaluation of a Determinant Using Properties

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Cofactors of aIj in |A|

The cofactor of an element in a determinant is the minor of multiplied by . It is usually denoted by. thus,

Cofactor of

Example:

Find the minors and cofactors of the elements of the second row in the determinant

Solution:

The elements of the second row are

Minor of (i.e., )

Minor of (i.e., )

And Minor of (i.e., )

The corresponding cofactors will be

And

Properties of Determinants:

We shall now discuss some of the properties of determinants. These properties will help us in expanding the determinants.

Property 1:

The value of a determinant remains unchanged if its rows and columns are interchanged.

Property 2:

If two rows (or columns) of a determinant are interchanged, then the value of the determinant changes in sign only.

Corollary:

If any row (or a column) of a determinant is passed over rows (or columns), then the resulting determinant is

Property 3:

If any two rows (or columns) of a determinant are identical then the value of the determinant is zero.

Property 4:

If each element of a row (or column) of a determinant is multiplied by the same constant say, , then the value of the determinant is multiplied by that constant .

Corollary:

If any two rows (or columns) of a determinant are proportional, then its value is zero.

Property 5:

If each element of a row (or of a column) of a determinant is expressed as the sum (or difference) of two or more terms, then the determinant can be expressed as the sum (or difference) of two or more determinants of the same order whose remaining rows (or columns) do not change.

Property 6:

The value of a determinant does not change, if to each element of a row (or a column) be added (or subtracted) the some multiples of the corresponding elements of one or more other rows (or columns)

Evaluation of a Determinant Using Properties:

Now we are in a position to evaluate a determinant easily by applying the aforesaid properties.

The purpose of simplification of a determinant is to make maximum possible zeroes in a row (or column) by using the above properties and then to expand the determinant by that row (or column). We denote and row by and respectively and and column by and respectively.

Example:

Show that where is a non-real cube root of unity.

Solution:

Add the sum of the and column to the column. We write this operation as

(On expanding by )

(Since is a non-real cube root of unity, therefore, )

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