1.Determinants important points class 12.pptx

Important notes for class 12 maths
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1. VINOD A V , PGT Mathematics, Jawahar Navodaya Vidyalaya , Thrissur -KERALA

2. To every square matrix of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A or , where is the element of A. . If , then determinant of is written as .   DETERMINANTS Only square matrices have determinants.

3. Determinant of a matrix of order one Let A = [ a ] be the matrix of order 1, then determinant of A is defined to be equal to a .  

4. Determinant of a matrix of order two Let us consider the following system of linear equations: On solving this system of equations for x and y, we get , provided . The number determines whether the values of and exist or not. The number is called the determinant of the matrix  

5. ie ., if be a matrix of order two. Then the determinant of is defined as .   Example Determinant of a matrix of order two

6. Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column).   There are six ways of expanding a determinant of order 3. Corresponding to each of three rows ( , and ) and three columns ( , and ) and each way gives the same value.   Determinant of a matrix of order 3

7. Consider the determinant of a square matrix , i.e .,   Expansion along first Row ( ) Step 1 Multiply first element of by and with the second order determinant obtained by deleting the elements of first row ( ) and first column ( ) of | A | as lies in and i.e .,  

8. Step 2 Multiply 2 nd element of by and with the second order determinant obtained by deleting the elements of first row ( ) and first column ( ) of | A | as lies in and i.e.,    

9. Step 3 Multiply 3 rd element of by and with the second order determinant obtained by deleting the elements of first row ( ) and first column ( ) of | A | as lies in and i.e.,    

10. Step 4 Now the expansion of determinant of A, that is, | A | written as sum of all three terms obtained in steps 1, 2 and 3 above is given by       Example Expanding along  

11. Q1. Find , if  

12. Q2. Find values of x for which .  

13. Q3. Find values of x for which .  

14. MINORS AND COFACTORS Minor of an element To each element of a square matrix, 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. Minor of an element of a square matrix is the determinant obtained by deleting its row and column in which element lies.   Minor of an element is denoted by .  

15. Examples: Minor of the element 7 in the matrix is 3   Minor of the element in the matrix is . .   Remark Minor of an element of a square matrix of order n ( n ≥ 2) is a determinant of order n – 1.

16. Cofactor of an element The cofactor of an element in a square matrix is the minor of multiplied by   It is usually denoted by ,   Thus, Cofactor of = =   Minors and Cofactors

17. Q. Find the cofactors of the elements , and of the determinant   Cofactor of   Cofactor of   Cofactor of  

18. Note:- The sum of the product of elements of any row (or column) with their corresponding cofactors is always equal to its Determinant value Now,    

19. If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero. Now,         .  

20. Example If is a cofactor of the element , of the determinant , then write the value of .