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Matrices, Introduction

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Matrices - Introduction, Matrix algebra has at least two advantages:, •Reduces complicated systems of equations to simple, expressions, •Adaptable to systematic method of mathematical treatment, and well suited to computers, Definition:, , A matrix is a set or group of numbers arranged in a square or, rectangular array enclosed by two brackets, , 1, , − 1, ,  4 2, − 3 0, , , , a b , c d , , 

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Matrices - Introduction, Properties:, •A specified number of rows and a specified number of, columns, •Two numbers (rows x columns) describe the dimensions, or size of the matrix., Examples:, 3x3 matrix, 2x4 matrix, 1x2 matrix, , 1 2 4 , 4 − 1 5 1 1, ,  , 3 3 3 0 0, , 3 − 3, , 3 2 , , 1, , − 1

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Matrices - Introduction, TYPES OF MATRICES, 1. Column matrix or vector:, The number of rows may be any integer but the number of, columns is always 1, , 1 ,  4,  , 2, , 1, − 3,  , ,  a11 ,  a21 ,  ,  , am1 

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Matrices - Introduction, TYPES OF MATRICES, 2. Row matrix or vector, Any number of columns but only one row, , 1, , 1 6, , a11, , a12, , 0, , 3 5 2, , a13  a1n 

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Matrices - Introduction, TYPES OF MATRICES, 3. Rectangular matrix, , Contains more than one element and number of rows is not, equal to the number of columns, , 1 1 , 3 7 , , , 7 − 7 , , , 7 6 , , 1 1 1 0 0 ,  2 0 3 3 0, , , , mn

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Matrices - Introduction, TYPES OF MATRICES, 4. Square matrix, The number of rows is equal to the number of columns, , (a square matrix A has an order of m), mxm, , 1 1, 3 0, , , , 1 1 1, 9 9 0, , , 6 6 1, , The principal or main diagonal of a square matrix is composed of all, elements aij for which i=j

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Matrices - Introduction, TYPES OF MATRICES, 5. Diagonal matrix, A square matrix where all the elements are zero except those on, the main diagonal, , 1 0 0, 0 2 0 , , , 0 0 1, i.e. aij =0 for all i = j, , aij = 0 for some or all i = j, , 3, 0, , 0, , 0, , 0, 3, 0, 0, , 0, 0, 5, 0, , 0, 0, 0, , 9

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Matrices - Introduction, TYPES OF MATRICES, 6. Unit or Identity matrix - I, , A diagonal matrix with ones on the main diagonal, , 1, 0, , 0, , 0, , 0, 1, 0, 0, , 0, 0, 1, 0, , 0, 0, 0, , 1, , i.e. aij =0 for all i = j, , aij = 1 for some or all i = j, , 1 0, 0 1 , , , , aij, 0, , , 0, aij 

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Matrices - Introduction, TYPES OF MATRICES, 7. Null (zero) matrix - 0, All elements in the matrix are zero, , 0 , 0 ,  , 0, , aij = 0, , 0 0 0 , 0 0 0 , , , 0 0 0, For all i,j

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Matrices - Introduction, TYPES OF MATRICES, 8. Triangular matrix, , A square matrix whose elements above or below the main, diagonal are all zero, , 1 0 0 ,  2 1 0, , , 5 2 3, , 1 0 0 ,  2 1 0, , , 5 2 3, , 1 8 9, 0 1 6 , , , 0 0 3

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Matrices - Introduction, TYPES OF MATRICES, 8a. Upper triangular matrix, A square matrix whose elements below the main, diagonal are all zero, , aij, , 0, 0, , , aij, aij, 0, , aij , , aij , aij , , i.e. aij = 0 for all i > j, , 1 8 7 , 0 1 8 , , , 0 0 3, , 1, 0, , 0, , 0, , 7, 1, 0, 0, , 4, 7, 7, 0, , 4, 4, 8, , 3

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Matrices - Introduction, TYPES OF MATRICES, 8b. Lower triangular matrix, A square matrix whose elements above the main diagonal are all, zero, , aij, , aij, aij, , , 0, aij, aij, , 0, , 0, aij , , i.e. aij = 0 for all i < j, , 1 0 0 ,  2 1 0, , , 5 2 3

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Matrices – Introduction, TYPES OF MATRICES, 9. Scalar matrix, , A diagonal matrix whose main diagonal elements are, equal to the same scalar, A scalar is defined as a single number or constant, , aij, , 0, 0, , , 0, aij, 0, , 0, , 0, aij , , i.e. aij = 0 for all i = j, aij = a for all i = j, , 1 0 0, 0 1 0 , , , 0 0 1, , 6, 0, , 0, , 0, , 0, 6, 0, 0, , 0, 0, 6, 0, , 0, 0, 0, , 6

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Matrices, Matrix Operations

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Matrices - Operations, EQUALITY OF MATRICES, Two matrices are said to be equal only when all, corresponding elements are equal, Therefore their size or dimensions are equal as well, , A=, , 1 0 0 ,  2 1 0, , , 5 2 3, , B=, , 1 0 0 ,  2 1 0, , , 5 2 3, , A=B

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Matrices - Operations, ADDITION AND SUBTRACTION OF MATRICES, , The sum or difference of two matrices, A and B of the same, size yields a matrix C of the same size, , cij = aij + bij, Matrices of different sizes cannot be added or subtracted