Determinants, Determinant of a Square Matrix, Expansion of a Determinant of Order 3, Minor of aij in |A|

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Determinant of a Square Matrix

  • With each square matrix of numbers (we associate) “determinant of the matrix”.

  • With the matrix ], we associate the determinant of order and with the only element . The value of the determinant is .

If

With the matrix , we associate the determinant and its value is defined to be

The determinant of a unit matrix is .

A square matrix whose determinant is zero, is called the singular matrix.

Example:

If , find

Solution:

Expansion of a Determinant of Order 3:

Which can be further expanded as

We notice that in the above method of expansion, each element of first row is multiplied by the second order determinant obtained by deleting the row and column in which the element lies.

Further, mark that the elements and have been assigned positive, negative and positive signs, respectively. In other words, they are assigned positive and negative signs, alternatively, starting with positive sign.

If the sum of the subscripts of the elements is an even number, we assign positive sign and if it is an odd number, then we assign negative sign. Therefore, has been assigned positive sign.

Note:

We can expand the determinant using any row or column. The value of the determinant will be the same whether we expand it using first row or first column or any row or column, taking into consideration rule of sign as explained above.

Example:

Expand the determinant, using the first row

Solution:

Minor of aIj in |A|

  • To each element of a determinant, a number called its minor is associated.

  • The minor of an element is the value of the determinant obtained by deleting the row and column containing the element.

  • Thus, the minor of an element in is the value of the determinant obtained by deleting the row and column of and is denoted by .

Example:

Find the minors of the elements of the determinant

Solution:

Let denote the minor of . Now, occurs in the row and column. Thus to find the minor of , we delete the row and column of .

The minor of is given by

Similarly, the minor of is given by

;

;

;

;

Similarly we can find , and .

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