1. Introduction Types of Matrices Operations
CHAPTER 2: MATRIX
Introduction
Types of Matrices
Operations

2. INTRODUCTION Definition 2.1
A matrix is a rectangular array of elements or entries aij involving m rows and n columns
Columns, n
Rows, m

3. INTRODUCTION Definition 2.2
2 matrices and are said to be equal iff m = r and n = s then A = B.
If aij for i = j, then the entries a11,a22,a33,… are called the diagonal of matrix A

4. Example 2.1
Find the values for the variables so that the matrices in each exercise are equal.

5. TYPES OF MATRICES
Square Matrix
Matrix with order n x n

6. TYPES OF MATRICES Diagonal Matrix
Matrix with order n x n with aij ≠ 0 and aij = 0 for i ≠ j

7. TYPES OF MATRICES Scalar Matrix
A diagonal matrix in which the diagonal elements are equal, aii = k and aij = 0 for i ≠ j where k is a scalar

8. TYPES OF MATRICES Identity Matrix
A diagonal matrix in which the diagonal elements are ‘1’, aii = 1 and aij ≠ 0 for i ≠ j

9. TYPES OF MATRICES Zero Matrix
A matrix which contains only zero elements, aij = 0

10. TYPES OF MATRICES Negative Matrix
A negative matrix of A =[aij] denoted by –A where -A =[-aij]

11. TYPES OF MATRICES Upper Triangular Matrix
If every elements below the diagonal is zero or aij = 0, i > j
DIAGONAL

12. TYPES OF MATRICES Lower Triangular Matrix
If every elements above the diagonal is zero or aij = 0, i < j
DIAGONAL

13. TYPES OF MATRICES Transpose of Matrix
If A =[aij] is an m x n matrix, then the transpose of A, AT =[aij]T is the n x m matrix defined by [aij] = [aji]T

14. TYPES OF MATRICES Properties Transposition Operation
Let A and B matrices and k, Then,

15. TYPES OF MATRICES
Example 1:
If and , find

16. TYPES OF MATRICES
Answer 1:

17. TYPES OF MATRICES Symmetric Matrix
If AT = A, where the elements obey the rule
aij = aji

18. TYPES OF MATRICES Skew Symmetric Matrix
If AT = - A, where the elements obey the rule
aij = - aji, so that the diagonal must contain zeroes.

19. TYPES OF MATRICES
Skew Symmetric Matrix

20. TYPES OF MATRICES Row Echelon Form (REF)
Matrix A is said to be in REF if it satisfies the following properties:
Rows consisting entirely zeroes occur at the bottom of the matrix.
For each row that doesn’t consist entirely of zeroes, the 1st nonzero is 1.
For each non zero row, number 1 appear to the right of the leading 1 of the previous row.

21. TYPES OF MATRICES
LEADING 1
ZERO ROW AT THE BOTTOM
LEADING 1

22. TYPES OF MATRICES Reduced Row Echelon Form (RREF)
Matrix A is said to be in REF if it satisfies the following properties:
Rows consisting entirely zeroes occur at the bottom of the matrix.
For each row that doesn’t consist entirely of zeroes, the 1st nonzero is 1.
For each non zero row, number 1 appear to the right of the leading 1 of the previous row.
If a column contains a leading 1, then all other entries in the column are zero

23. TYPES OF MATRICES
LEADING 1
ZERO ROW AT THE BOTTOM
LEADING 1

24. OPERATIONS OF MATRICES
Example 2: Given , find:

25. OPERATIONS OF MATRICES