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

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

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

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

TYPES OF MATRICES Square Matrix Matrix with order n x n

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

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

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

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

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

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

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

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

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

TYPES OF MATRICES Example 1: If and , find

TYPES OF MATRICES Answer 1:

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

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

TYPES OF MATRICES Skew Symmetric Matrix

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.

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

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

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

OPERATIONS OF MATRICES Example 2: Given , find:

OPERATIONS OF MATRICES