Every square matrix is associated with a unique number called the determinant of the matrix. In this lesson, we will learn various properties of determinants and also evaluate determinants by different methods.
After seen this vides ,you will be able to :
- define determinant of a square matrix;
- define the minor and the cofactor of an element of a matrix;
- find the minor and the cofactor of an element of a matrix;
- find the value of a given determinant of order not exceeding
- state the properties of determinants; evaluate a given determinant of order not exceeding 3 by using expansion method;
EXPECTED BACKGROUND KNOWLEDGE
- Knowledge of solution of equations
- Knowledge of number system including complex number
- Four fundamental operations on numbers and expressions
DETERMINANT OF ORDER 2
Let us consider the following system of linear equations:
On solving this system of equations for x and y, we get
DETERMINANT OF ORDER 3
EXPANSION OF A DETERMINANT OF ORDER 3
In Section 4.4, we have written
Example 21.5 Expand the determinant, using the first row
Example 21.6 Expand the determinant, by using the second column
MINORS AND COFACTORS
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 aij in |A| is the value of the determinant obtained by deleting the i th row and jth column of |A| and is denoted by Mij. For example, minor of 3 in the determinant.
Example 21.7 Find the minors of the elements of the determinant
Let Mij denote the minor of aij. Now, a11 occurs in the 1st row and 1st column. Thus to find the minor of a11, we delete the 1st row and 1st column of |A|. The minor M11 of A11 is given by
Example 21.8 Find the cofactors of the elements a11, a12,and a21 of the determinant
Example 21.9 Find the minors and cofactors of the elements of the second row in the determinant
CHECK YOUR PROGRESS 21.2
1. Find the minors and cofactors of the elements of the second row of the determinant
2. Find the minors and cofactors of the elements of the third column of the determinat
3. Evaluate each of the following determinants using cofactors:
4. Solve for x, the following equations:
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.
Expanding the determinant by first row, we have
can be expressed as the sum of the determinants of the same order.
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 1st, 2nd and 3rd row by R1, R2 , and R3 respectively and 1st,2nd and 3rd column by C1 , C2 and C3 respectively.
CHECK YOUR PROGRESS 21.3
Application of Determinants
Determinant is used to find area of a triangle.