nhaa/imk/sem120162017/eqt101/rk12/32 CHAPTER 2: MATRICES Introduction Types of Matrices Operation Determinant Inverse nhaa/imk/sem120162017/eqt101/rk12/32 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 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 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 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 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 Example Find the values for the variables so that the matrices in each exercise are equal. nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Square Matrix Matrix with order n x n nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Diagonal Matrix Matrix with order n x n with aij ≠ 0 and aij = 0 for i ≠ j nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 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 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Identity Matrix A diagonal matrix in which the diagonal elements are ‘1’, aii = 1 and aij ≠ 0 for i ≠ j nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Zero Matrix A matrix which contains only zero elements, aij = 0 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Negative Matrix A negative matrix of A =[aij] denoted by –A where -A =[-aij] nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Upper Triangular Matrix If every elements below the diagonal is zero or aij = 0, i > j DIAGONAL nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Lower Triangular Matrix If every elements above the diagonal is zero or aij = 0, i < j DIAGONAL nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 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 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Properties Transposition Operation Let A and B matrices and k, . Then, nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Example 1: If and , find nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Symmetric Matrix If AT = A, where the elements obey the rule aij = aji nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Skew Symmetric Matrix If AT = - A, where the elements obey the rule aij = - aji, so that the diagonal must contain zeroes. nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Skew Symmetric Matrix nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 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. nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES LEADING 1 ZEROS ROW AT THE BOTTOM LEADING 1 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES Reduced Row Echelon Form (RREF) Matrix A is said to be in RREF 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 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 TYPES OF MATRICES LEADING 1 ZEROS ROW AT THE BOTTOM LEADING 1 nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Definition 2.15 Let and are matrices of order mxn. Matrices C=A+B is defined by which is and Two matrices A and B will be said conformable for addition only if they are both of the same order. nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Definition 2.16 Let and are matrices of order mxn. Matrices C=A-B is defined by which is and Two matrices A and B will be said conformable for subtraction only if they are both of the same order. nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Properties of Addition and Subtraction If then nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Definition 2.17 Let is an mxn matrix, , then the scalar multiplication is denoted where nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Properties of Scalar Multiplication If and then nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Definition 2.18 Suppose A is an mxn matrix and B is a pxq matrix. For he product AB to exist, it must be that n=p, that is the number of columns in A must be the same as the number of rows in B. nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Properties of Matrix Multiplication If and C are matrices, I, identity matrix and , zero matrix, then nhaa/imk/sem120162017/eqt101/rk12/32

OPERATIONS OF MATRICES Example 2: Given , find: nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 DETERMINANT : 2X2 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 nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 DETERMINANT : 2X2 Example If and find det(A) and det (B). Solution: nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 DETERMINANT : 3x3 Definition 2.20 Given is a 3x3 matrix, then the determinant is given by: nhaa/imk/sem120162017/eqt101/rk12/32

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: nhaa/imk/sem120162017/eqt101/rk12/32

DETERMINANT : 3x3 (minors & cofactors) Definition 2.22 Cofactor of aij : For 3x3 matrix : We can conclude that : Even = 1, Odd = -1 nhaa/imk/sem120162017/eqt101/rk12/32

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 nhaa/imk/sem120162017/eqt101/rk12/32

DETERMINANT : 3x3 (minors & cofactors) Therefore : Cofactor of aij = nhaa/imk/sem120162017/eqt101/rk12/32

DETERMINANT : 3x3 (minors & cofactors) Exercise 1: Find [Cij]: nhaa/imk/sem120162017/eqt101/rk12/32

DETERMINANT : 3x3 (minors & cofactors) Example Find determinant of A: nhaa/imk/sem120162017/eqt101/rk12/32

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) nhaa/imk/sem120162017/eqt101/rk12/32

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. nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 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. nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 ADJOINT Example Find the adjoint of A: nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 ADJOINT Example Find the adjoint of B: nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 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. nhaa/imk/sem120162017/eqt101/rk12/32

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.

nhaa/imk/sem120162017/eqt101/rk12/32 INVERSE : 2X2 Example Find the inverse for the given matrix: nhaa/imk/sem120162017/eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32 INVERSE : 2X2 Example If , show that is the inverse of A. Solution Use definition if B is the inverse of A nhaa/imk/sem120162017/eqt101/rk12/32

INVERSE FOR 3X3 COFACTOR METHOD ELEMENTARY ROW OPERATION (ERO) nhaa/imk/sem120162017/eqt101/rk12/32

Inverse using the Cofactor Method Theorem 2 If A is nxn matrix, |A|≠0, then A-1 is defined by: nhaa/imk/sem120162017/eqt101/rk12/32

Inverse using the Cofactor Method Example Find the inverse of each matrix using the Cofactor Method: nhaa/imk/sem120162017/eqt101/rk12/32

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/sem120162017/eqt101/rk12/32

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (i) : interchange the elements between ith row and jth row Example nhaa/imk/sem120162017/eqt101/rk12/32

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (ii) : multiply ith row by a nonzero scalar, k Example NEW R1 nhaa/imk/sem120162017/eqt101/rk12/32

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 nhaa/imk/sem120162017/eqt101/rk12/32

Inverse using Elementary Row Operations (ERO) Method of solving using ERO Step 1: write A in augmented form Step 2: use characteristics i,ii or iii to reduce A nhaa/imk/sem120162017/eqt101/rk12/32

Inverse using Elementary Row Operations (ERO) Example Find the inverse of each matrix using the Elementary Row Operations (ERO) nhaa/imk/sem120162017/eqt101/rk12/32