Matrices and their applications
- 1. MATRICES Presented by: Farhan Tariq Ali Hassan Zargham Abbas
- 2. WHAT IS A MATRIX ? • A matrix is a set of elements arranged in rows and columns or rectangular array of number enclosed in brackets. 2 3 7 1 -1 5 A= B= 1 2 3 4 4 6 Both A and B are examples of matrices. • Generally they are represented in parenthesis
- 3. ORDER OF MATRIX • Number of rows and columns which a matrix has is called order of matrix. Rows are written horizontally and columns are vertically. 1 5 8 2 6 8 3 7 5 A= Matrix A has 3 rows and 3 columns, so it is matrix of order 3x3.Rows are represented with “m” and columns are with “n”.Generally, m x n is called order of matrix. Similarly, 2 5 3 7 4 9 3 4 7 8 4 3 1 0 5 6 4 5 B= C= B is a matrix of order 3x2 while C is a matrix of order 3x4.
- 4. TYPES OF MATRICES • Square matrix When m=n, a matrix is called square matrix. Example A= where 2,5,7 are called Diagonal elements. • Trace of any matrix is found by adding the elements in diagonal. Trace of A is 2+5+7=14. • Equal matrices Two matrices are said to be equal if each element of first matrix is equal to the corresponding element of second matrix. A= B= A and B are only said to be equal in this type of state. 2 3 7 2 5 8 3 6 7 2 3 3 4 2 3 3 4
- 5. • Zero Matrix If every element of a matrix is zero, then it is called zero or null matrix. N= • Upper triangular matrix is a matrix whose all elements below the diagonal elements are zero. F= 1 2 3 0 5 6 0 0 1 • Lower triangular matrix is a matrix whose all elements upper the diagonal elements are zero. D= 1 0 0 2 5 0 3 6 7 0 0 0 0
- 6. • Identity Matrix Both upper and lower triangular matrix whose all diagonal elements are I= ‘1’ is called identity matrix. • Transpose of matrix It is obtained by interchanging the rows and columns of a matrix. If A = • Symmetric Matrix A matrix A such that A=AT is called symmetric matrix. If At =-A, then it is called a skew symmetric matrix. 1 0 0 0 1 0 0 0 1 2 9 3 4 2 3 9 4 Transpose of A=
- 7. • Orthogonal Matrix A matrix A is called to be orthogonal only if AAT =ATA= Identity matrix. We can say that orthogonal means AT=A-1 . • Properties of matrices 1. (AB)-1=B-1A-1 2. (At)t =A and (kA)t=k At 3. (A+B)T=At + Bt 4. (AB)t=Bt At
- 8. DETERMINANTS • Determinant of 2x2 matrix Consider a matrix A = , Then its determinant can be founded as follows: |A|= 2 3 |A|=2x6 – 4x3 =12 – 12= 0 • Determinant Of 3x3 Matrix 3 2 1 2 4 7 = 3 - 2 + 1 =3(12-14) – 2(6-7) + 1(4-4)= -4 1 2 3 2 3 4 6 4 6 4 7 2 3 2 7 1 3 2 4 1 2
- 9. • Properties of Determinants • |At|=|A| , A matrix and its transpose has same determinant. • |A|= 0 ,If two rows or columns are same. • K |A|= k det A • Det A= -|A|, if we interchange one or two rows and columns. • |A|=0, If all elements of any row or column are zero. • |AB|= |A||B| • If matrix is upper or lower triangular, determinant can be found by multiplying the elements of diagonals.
- 10. Echelon Form • A matrix is in row echelon form when it satisfies the following conditions. The first non-zero element in each row, called the leading entry, is 1. Each leading entry is in a column to the right of the leading entry in the previous row. A = 1 2 3 0 0 4 0 0 0
- 11. • Reducing a matrix in row echelon form To reduce any square matrix into row echelon form , utilize these steps in order to get your desired row echelon form. 1st step 2ND step 1 0 0 0 1 0 0 0 1 3rd step 4th step 4th step 5th step 6th step