1. CHAPTER 2 MATRICES
Determinant
Inverse

2. DETERMINANT : 2X2 Definition 2.19
Determinant of matrix A is defined by det(A) or |A|
Definition 2.19
If is a 2x2 matrix, then the determinant is
given by

3. INVERSE : 2X2 Definition If , then is the inverse of A where Theorem 1
Matrix A in invertible if and only if
If , then A doesn’t have an inverse.

4. INVERSE : 2X2
Example 1
Find the inverse for the given matrix:

5. INVERSE : 2X2
Example 1
Find the inverse for each matrix:

6. INVERSE : 2X2
Example 2
If , show that is the inverse of A.

7. DETERMINANT : 3x3 Definition 2.20 Given is a 3x3 matrix, then the
determinant is given by:

8. DETERMINANT : 3x3 (minors & cofactors)
Definition 2.23
If Cij is cofactor of matrix A, then det(A) can be obtained by:
Expanding along the ith row:
Expanding along the jth column:

9. DETERMINANT : 3x3 (minors & cofactors)
Definition 2.22
Cofactor of aij :
For 3x3 matrix :
We can conclude that : Even = 1, Odd = -1

10. DETERMINANT : 3x3 (minors & cofactors)
Definition 2.21
Let n ≥ 2 and A = [ aij ]nxn . Matrices (n -1)x (n -1) submatrix of A is obtained by deleting the ith row and jth column of A, denoted by Mij .
Minor of

11. DETERMINANT : 3x3 (minors & cofactors)
Therefore :
Cofactor of aij =

12. DETERMINANT : 3x3 (minors & cofactors)
Exercise 1:
Find determinant of A:

13. DETERMINANT : 3x3 (minors & cofactors)
Answer 1
Minor=
Cofactor =

14. DETERMINANT : 3x3 (minors & cofactors)
Exercise 2:
Find determinant of A:
ANSWER:

15. DETERMINANT : PROPERTIES
Suppose A is nxn matrix and k is a scalar. Suppose the matrix B is obtained by multiplying a single row or column of A by k. Then det(B) = k det(A)
If matrix A is multiplied by k, that is every element in the matrix is multiplied by k, then det(kA) = kn det(A)
If B is obtained from A by interchanging 2 rows or 2 columns, then det(B) = - det(A)

16. DETERMINANT : PROPERTIES
Adding or subtraction a multiple of one row(column) to the other row(column) leaves the determinant unchanged
If A and B are 2 square matrices such that AB exists, then, det(AB) = det(A) det(B)
If 2 rows or 2 columns of a matrix are equal, the determinant of the matrix is zero.

17. ADJOINT
Definition 2.24
Let A is an nxn matrix, then the transpose of the matrix of cofactors A is called the matrix adjoint to A.

18. ADJOINT
Exercise 3:
Find the inverse of A:

19. ADJOINT

20. ADJOINT
Answer 3:
Cofactor =
adj(A) =

21. ADJOINT
Exercise 4:
Find inverse of B:

22. ADJOINT
Answer 4:
Cofactor =
adj(B) =

23. INVERSE
Definition 2.25
If A is a square matrix of order n and if there exists a matrix A-1 such that
then A-1 is called the inverse of A.

24. ELEMENTARY ROW OPERATION
INVERSE FOR 3X3
COFACTOR METHOD
ELEMENTARY ROW OPERATION
(ERO)

25. Inverse using Cofactor Method
Theorem 2
If A is nxn matrix, |A|≠0, then A-1 is defined by:

26. INVERSE
Exercise 7:
Find the inverse of each matrix using the Cofactor Method:

27. INVERSE Answer 7: Solution for (a): Step 1: find cofactor of A
Step 2: find det(A)

28. INVERSE
Step 3: find adj(A)
Step 4: find the inverse of A

29. Inverse using Elementary Row Operations (ERO)
Theorem 3
Let A and I both be nxn matrices, the augmented matrix
may be reduced to by using elementary row operation (ERO)

30. Inverse using Elementary Row Operations (ERO)
Characteristics of ERO
(i) : interchange the elements between ith row and jth row
Example

31. Inverse using Elementary Row Operations (ERO)
Characteristics of ERO
(ii) : multiply ith row by a nonzero scalar, k
Example
NEW R1

32. Inverse using Elementary Row Operations (ERO)
Characteristics of ERO
(iii) : add or subtract ith row to a constant multiple jth row by a nonzero scalar, k
Example
NEW R1

33. Inverse using Elementary Row Operations (ERO)
Exercise 8:
Find the inverse of each matrix using the Elementary Row Operations (ERO)