1. Introduction and Definitions
MATRIX
Introduction and Definitions

5. Types of matrices Square matrix Diagonal matrix Scalar matrix
Identity matrix (ones “1” on the main diagonal and zeros “0”everywhere else)
Zero matrix (contains only zero elements)
Negative matrix

6. Reduced row echelon form (RREF)
Upper and lower triangular matrix (called upper-triangular if every element below the leading diagonal is zero and called lower-triangular matrix if every element above the leading diagonal is zero)
Transpose matrix
Symmetric matrix
Skew symmetric matrix
Row echelon form (REF)
Reduced row echelon form (RREF)

8. Row Echelon Form (REF)

10. Reduced Row Echelon Form (RREF)

19. Determinants

20. Second Order Determinants
Third Order Determinants
Higher Order Determinants
Minors and Cofactors

21. Second Order Determinants

23. Third Order Determinants

25. Note: Methods used to evaluate the determinant above is limited to only 2× 2
and 3×3 matrices. Matrices with higher order can be solved by using minor
and cofactor methods.

26. Higher Order Determinants

32. Adjoint

35. The Inverse of a
Square Matrix

36. Inverse of a Matrix
Finding the inverse of a 2x2 Matrix
Finding the inverse of a 3x3 or Higher order Matrix
By using cofactor method
By using elementary row operation

39. Finding the inverse of a 2x2 Matrix

42. ELEMENTARY ROW OPERATION
INVERSE FOR 3X3
ELEMENTARY ROW OPERATION
(ERO)
COFACTOR METHOD
NHAA/IMK/UNIMAP

43. Finding the inverse of a 3x3 or Higher order Matrix by using cofactor method

44. Find the inverse of each matrix using the Cofactor Method:
Exercise:
Find the inverse of each matrix using the Cofactor Method:
NHAA/IMK/UNIMAP

45. Step 1: find the cofactor of A
Answer :
Solution for (a):
Step 1: find the cofactor of A
Step 2: find adj(A)
NHAA/IMK/UNIMAP

46. Step 4: find the inverse of A
Step 3: find det(A)
Step 4: find the inverse of A
NHAA/IMK/UNIMAP

47. Finding the inverse of a 3x3 or Higher order Matrix by using Elementary Row Operation

48. 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)
NHAA/IMK/UNIMAP

49. Inverse using Elementary Row Operations (ERO)
Characteristics of ERO
(i) : interchange the elements between ith row and jth row
Example
NHAA/IMK/UNIMAP

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

51. 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
NHAA/IMK/UNIMAP

55. Types of Solutions to system
Linear Equations

56. There are 3 possible solutions:
3 TYPES OF SOLUTIONS
A SYSTEM WITH UNIQUE SOLUTION
A SYSTEM WITH INFINITELY MANY SOLUTIONS
A SYSTEM WITH NO SOLUTION
NHAA/IMK/UNIMAP

57. A System with Unique Solution
Consider the system:
Augmented matrix:
The system has unique solution:
NHAA/IMK/UNIMAP

58. A System with Infinitely Many Solutions
Consider the system:
Augmented matrix:
The system has many solutions: let where s is called a free variable. Then,

59. A system with No Solution
Consider the system:
Augmented matrix:
The system has no solution, since coefficient of is ‘0’.
NHAA/IMK/UNIMAP

60. Solving systems of Equations
Inverses of Matrices
Gaussian Elimination and Gauss-Jordan Elimination
Cramer’s Rule

62. Inverses of Matrices

63. Inverses of Matrices

64. Gauss-Jordan Elimination
Gaussian Elimination
and
Gauss-Jordan Elimination

68. Cramer’s Rule

69. Cramer’s Rule

70. Cramer’s Rule

71. Cramer’s Rule

72. Cramer’s Rule

73. Thank You