Mathematics for Computing CLO 3.1 PDF

LSM 1013 – Mathematics for Computing CLO 3.1 - Define a matrix and identify the difference between various types of matrices, such as row, column, square, diagonal, scalar, identity, vector, and zero matrices. Determine the transpose for any given matrix. 3.1.1 – Introduction...

LSM 1013 – Mathematics for Computing CLO 3.1 - Define a matrix and identify the difference between various types of matrices, such as row, column, square, diagonal, scalar, identity, vector, and zero matrices. Determine the transpose for any given matrix. 3.1.1 – Introduction 2 A matrix (plural matrices) is a rectangular array of numbers written within brackets. Example 1: Each number in a matrix is called an element of the matrix. How many elements are there in Matrix A and B? If a matrix has m rows and n columns, it is called an m x n matrix (read “m by n matrix”). The expression is called the size of the matrix, and the numbers m and n are called the dimensions of the matrix. Note: The number of rows is always given first. Referring to Example 1, A is a 2 x 3 matrix and B is a 4 x 3 matrix. 3.1.1 – Introduction 3 A matrix with n rows and n columns is called a square matrix of order n. A matrix with only one column is called a column matrix. A matrix with only one row is called a row matrix. Example 2: Note: If a matrix has only one row or only one column it is called a vector. Hence, a column matrix s also called a column vector and a row matrix is called a row vector. 3.1.1 – Introduction 4 The position of an element in a matrix is the row and column containing the element. This is usually denoted using double subscript notation aij, where i is the row and j is the column containing the element aij, Note: a12 is read “a sub one two,” not “a sub twelve.” The elements a11 = 1, a22 = 0, and a33 = 4 make up the principal diagonal of A. In general, the principal diagonal of a matrix A consists of the elements aii, i 1, 2,... n. Example Find the principal diagonal of matrix B. 3: B= 3.1.1 – Introduction 5 A matrix with elements that are all 0’s is called a zero matrix. Example 3: A square matrix with non-diagonal elements equal to zero is called a diagonal matrix. Example 4: [ 1 0 0 2 ][ 1 0 0 0 2 0 0 0 3 ] An identity matrix is a diagonal matrix with all its diagonal elements equal to 1 , and zeroes everywhere else. I2 = I3 = Example I1 = 5: 3.1.1 – Introduction 6 A scalar matrix is a square matrix having a constant value as every element of its principal diagonal, and all other elements are equal to zero, i.e. A= where the principal diagonal elements are all equal to the same numeric value of 'a', and all other elements of the matrix are equal to zero. The scalar matrix is derived from an identity matrix, where the product of the identity matrix with a constant value, gives the scalar matrix. B= Example 6:  7 3.1.2 – Transpose of a Matrix The transpose, denoted by AT (where A is the original matrix), is a new matrix created by flipping the rows and columns. Essentially, the elements that were originally in each row become the corresponding elements in each column of the transpose. That is, If The elements swap positions across the main diagonal (running from top left to bottom right) Example 1: Find the transpose of the matrix Answer: Answer: Example 3: Find the transpose of a scalar matrix. Exercise 3.1 8 1. Identifying Matrix Types a) Given the matrix determine: Is A a square matrix? (Yes/No) Is A a row matrix? (Yes/No) Is A a column matrix? (Yes/No) b) Given the matrix , determine: Is B a diagonal matrix? (Yes/No) Is B a scalar matrix? (Yes/No) Is B an identity matrix? (Yes/No) Exercise 3.1 9 2. Create a 3 x 3 scalar matrix where all elements are equal to 7. 3. Find the transpose of a 2 x 2 identity matrix. 4. Find the transpose of the following matrices: a) b) [ ] 1 0 0 c) 0 2 0 0 0 3 Thank You