Matrices and Determinants - Matrices and Determinants...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of 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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 .
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern