Matrices and Determinants

# Matrices and Determinants - Matrices and Determinants...

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Matrices and Determinants Advanced Level Pure Mathematics Advanced Level Pure Mathematics Chapter 8 Matrices and Determinants 8.1 INTRODUCTION : MATRIX / MATRICES 2 8.2 SOME SPECIAL MATRIX 3 8.3 ARITHMETRICS OF MATRICES 4 8.4 INVERSE OF A SQUARE MATRIX 16 8.5 DETERMINANTS 19 8.6 PROPERTIES OF DETERMINANTS 21 8.7 INVERSE OF SQUARE MATRIX BY DETERMINANTS 27 Prepared by K. F. Ngai Page 1 8

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Matrices and Determinants Advanced Level Pure Mathematics 8.1 INTRODUCTION : MATRIX / MATRICES 1. A rectangular array of m × n numbers arranged in the form a a a a a a a a a n n m m mn 11 12 1 21 22 2 1 2 is called an m × n matrix . e.g. 2 3 4 1 8 5 - is a 2 × 3 matrix. e.g. 2 7 3 - is a 3 × 1 matrix. 2. If a matrix has m rows and n columns , it is said to be order m × n. e.g. 2 0 3 6 3 4 7 0 1 9 2 5 is a matrix of order 3 × 4. e.g. 1 0 2 2 1 5 1 3 0 - - is a matrix of order 3. 3. [ ] a a a n 1 2 is called a row matrix or row vector . 4. b b b n 1 2 is called a column matrix or column vector . e.g. 2 7 3 - is a column vector of order 3 × 1. e.g. [ ] - - - 2 3 4 is a row vector of order 1 × 3. 5. If all elements are real, the matrix is called a real matrix. Prepared by K. F. Ngai Page 2
Matrices and Determinants Advanced Level Pure Mathematics 6. a a a a a a a a a n n n n nn 11 12 1 21 22 2 1 2 is called a square matrix of order n. And a a a nn 11 22 , , , is called the principal diagonal. e.g. 3 9 0 2 - is a square matrix of order 2. 7. Notation : [ ] ( 29 a a A ij m n ij m n × × , , , ... 8.2 SOME SPECIAL MATRIX. Def.8.1 If all the elements are zero, the matrix is called a zero matrix or null matrix, denoted by O m n × . e.g. 0 0 0 0 is a 2 × 2 zero matrix, and denoted by O 2 . Def.8.2 Let [ ] A a ij n n = × be a square matrix. (i) If a ij = 0 for all i, j, then A is called a zero matrix. (ii) If a ij = 0 for all i<j, then A is called a lower triangular matrix . (iii) If a ij = 0 for all i>j, then A is called a upper triangular matrix . a a a a a a n n nn 11 21 22 1 2 0 0 0 0 0 a a a a a n nn 11 12 1 22 0 0 0 0 0 i.e. Lower triangular matrix Upper triangular matrix e.g. 1 0 0 2 1 0 1 0 4 - is a lower triangular matrix. e.g. 2 3 0 5 - is an upper triangular matrix. Prepared by K. F. Ngai Page 3

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Matrices and Determinants Advanced Level Pure Mathematics Def.8.3 Let [ ] A a ij n n = × be a square matrix. If a ij = 0 for all i j , then A is called a diagonal matrix .
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