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Tutorial 1 Matrices

D
Dr. Mehar Chand

The document defines and describes various types of matrices, including square, row, column, diagonal, scalar, identity, null, upper triangular, lower triangular, and triangular matrices. It also defines equality of matrices. The types of matrices are defined based on their structure, with square matrices having the same number of rows and columns, row and column matrices having only one row or column, and triangular matrices having either all elements above or below the diagonal equal to zero.

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Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Tutorial-I: Matrices Basic Mathematics: Matrices Dr. Mehar Chand Department of Mathematics Baba Farid College, Bathinda-151001 April 21, 2020 Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Contents 1 Introduction 2 Types of Matrices Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix 3 Equality of matrix 4 Full video Lecture Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Introduction The Matrices play very important role in Science, Engineer- ing, Business and Economics. There are many applications in various topics such as solving simultaneous equations, linear programming, input-output analysis etc. Definition (Matrix) A matrix of order m × n (can be read as m by n) is an ordered rectangular arrangement of m, n numbers or functions consisting of m rows and n columns and is defined as: A = [aij ]m×n Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Square Matrix Definition (Square Matrix) A matrix A = [aij ]m×n is called a square matrix if m = n i.e. the number of rows are equal to columns. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Row Matrix Definition (Row Matrix) A matrix A = [aij ]m×n is called a row matrix if m = 1 i.e. the matrix has only one row. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Column Matrix Definition (Column Matrix) A matrix A = [aij ]m×n is called a column matrix if n = 1 i.e. the matrix has only one column Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Diagonal Matrix Definition (Diagonal Matrix) A square matrix A = [aij ]m×n is called a Diagonal Matrix if aij = 0 for i = j and aij = 0 for i = j . a11, a22, ... are called diagonal elements of square matrix. The diagonal elements are said to constitute principal diagonal or simply diagonal of the matrix. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Scalar Matrix Definition (Scalar Matrix) A matrix A = [aij ]m×n is called a Scalar matrix if aij = a for i = j and aij = 0 for i = j. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Identity Matrix Definition (Identity Matrix) A matrix A = [aij ]m×n is called a Identity matrix if aij = 1 for i = j and aij = 0 for i = j. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Null (or Zero) Matrix Definition (Null (or Zero) Matrix) A matrix A = [aij ]m×n is called a Null (or Zero) Matrix if each elements of the matrix are zero. The zero matrix of order m × n is usually denoted by Om×n or O Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Upper triangular matrix Definition (Upper triangular matrix) A square matrix A = [aij ]n×n is called a Upper triangular matrix if aij = 0 for i > j. In other words we can say that a square matrix is an upper triangular matrix is all the elements below the principal diagonal are zero. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Lower triangular matrix Definition (Lower triangular matrix) A square matrix A = [aij ]n×n is called a Upper triangular matrix if aij = 0 for i < j. In other words we can say that a square matrix is an Lower triangular matrix is all the elements above the principal diagonal are zero. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Triangular matrix Definition (Triangular matrix) A square matrix A = [aij ]n×n is called a triangular matrix if it is either upper triangular matrix or lower triangular matrix. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Equality of matrices Definition (Equal of matrix) Two matrices A = [aij ]m×n and B = [bij ]m×n are said to be equal if they are of same order and their corresponding elements are equal. Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Full video Lecture To see full video lecture visit the below link https://youtu.be/hEId2ZuD4Qc Dr. Mehar Chand Properties Associated with Sphere
Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Thank you Dr. Mehar Chand Properties Associated with Sphere

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Tutorial 1 Matrices

  • 1. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Tutorial-I: Matrices Basic Mathematics: Matrices Dr. Mehar Chand Department of Mathematics Baba Farid College, Bathinda-151001 April 21, 2020 Dr. Mehar Chand Properties Associated with Sphere
  • 2. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Contents 1 Introduction 2 Types of Matrices Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix 3 Equality of matrix 4 Full video Lecture Dr. Mehar Chand Properties Associated with Sphere
  • 3. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Introduction The Matrices play very important role in Science, Engineer- ing, Business and Economics. There are many applications in various topics such as solving simultaneous equations, linear programming, input-output analysis etc. Definition (Matrix) A matrix of order m × n (can be read as m by n) is an ordered rectangular arrangement of m, n numbers or functions consisting of m rows and n columns and is defined as: A = [aij ]m×n Dr. Mehar Chand Properties Associated with Sphere
  • 4. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Square Matrix Definition (Square Matrix) A matrix A = [aij ]m×n is called a square matrix if m = n i.e. the number of rows are equal to columns. Dr. Mehar Chand Properties Associated with Sphere
  • 5. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Row Matrix Definition (Row Matrix) A matrix A = [aij ]m×n is called a row matrix if m = 1 i.e. the matrix has only one row. Dr. Mehar Chand Properties Associated with Sphere
  • 6. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Column Matrix Definition (Column Matrix) A matrix A = [aij ]m×n is called a column matrix if n = 1 i.e. the matrix has only one column Dr. Mehar Chand Properties Associated with Sphere
  • 7. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Diagonal Matrix Definition (Diagonal Matrix) A square matrix A = [aij ]m×n is called a Diagonal Matrix if aij = 0 for i = j and aij = 0 for i = j . a11, a22, ... are called diagonal elements of square matrix. The diagonal elements are said to constitute principal diagonal or simply diagonal of the matrix. Dr. Mehar Chand Properties Associated with Sphere
  • 8. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Scalar Matrix Definition (Scalar Matrix) A matrix A = [aij ]m×n is called a Scalar matrix if aij = a for i = j and aij = 0 for i = j. Dr. Mehar Chand Properties Associated with Sphere
  • 9. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Identity Matrix Definition (Identity Matrix) A matrix A = [aij ]m×n is called a Identity matrix if aij = 1 for i = j and aij = 0 for i = j. Dr. Mehar Chand Properties Associated with Sphere
  • 10. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Null (or Zero) Matrix Definition (Null (or Zero) Matrix) A matrix A = [aij ]m×n is called a Null (or Zero) Matrix if each elements of the matrix are zero. The zero matrix of order m × n is usually denoted by Om×n or O Dr. Mehar Chand Properties Associated with Sphere
  • 11. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Upper triangular matrix Definition (Upper triangular matrix) A square matrix A = [aij ]n×n is called a Upper triangular matrix if aij = 0 for i > j. In other words we can say that a square matrix is an upper triangular matrix is all the elements below the principal diagonal are zero. Dr. Mehar Chand Properties Associated with Sphere
  • 12. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Lower triangular matrix Definition (Lower triangular matrix) A square matrix A = [aij ]n×n is called a Upper triangular matrix if aij = 0 for i < j. In other words we can say that a square matrix is an Lower triangular matrix is all the elements above the principal diagonal are zero. Dr. Mehar Chand Properties Associated with Sphere
  • 13. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Triangular matrix Definition (Triangular matrix) A square matrix A = [aij ]n×n is called a triangular matrix if it is either upper triangular matrix or lower triangular matrix. Dr. Mehar Chand Properties Associated with Sphere
  • 14. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Equality of matrices Definition (Equal of matrix) Two matrices A = [aij ]m×n and B = [bij ]m×n are said to be equal if they are of same order and their corresponding elements are equal. Dr. Mehar Chand Properties Associated with Sphere
  • 15. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Full video Lecture To see full video lecture visit the below link https://youtu.be/hEId2ZuD4Qc Dr. Mehar Chand Properties Associated with Sphere
  • 16. Properties Associated with Sphere Dr. Mehar Chand Introduction Types Square Matrix Row Matrix Column Matrix Diagonal Matrix Scalar Matrix Identity Matrix Null Matrix Upper triangular matrix Lower triangular matrix Triangular matrix Equality of matrix Video Thank you Dr. Mehar Chand Properties Associated with Sphere