Presentation on matrix

presentation on Matrix and its application

1. Presented by Masuda Mahbub Nahin Mahfuz seam Saddique Muhammad Takbir Dakhin Dhaka school of Economics University of Dhaka Presentation on Matrix and it`s aplication

2. Outline Definition of a Matrix Operations of Matrices Determinants Inverse of a Matrix Linear System Unique properties of matrix Uses of matrices 1

3. Matrix (Basic Definitions) Matrices are the rectangular agreement of numbers, expressions, symbols which are arranged in columns and rows. 2

4. Operations with Matrices (Sum,Difference) If A and B have the same dimensions, then their sum, A + B, is obtained by adding corresponding entries. In symbols, (A + B) ij = a ij + b ij . If A and B have the same dimensions, then their difference, A − B, is obtained by subtracting corresponding entries. In symbols, (A - B) ij = a ij - b ij . 3

5. Operations with Matrices (Scalar Multiple) Example: If A is a matrix and r is a number (sometimes called a scalar in this context), then the scalar multiple, rA, is obtained by multiplying every entry in A by r. In symbols, (rA) ij = ra ij . 4

6. Operations with Matrices (Product) If A has dimensions k × m and B has dimensions m × n, then the product AB is defined, and has dimensions k × n. The entry (AB) ij is obtained by multiplying row i of A by column j of B, which is done by multiplying corresponding entries together and then adding the results i.e., 5

7. Laws of Matrix Algebra The matrix addition, subtraction, scalar multiplication and matrix multiplication, have the following properties. 6

8. Operations with Matrices (Transpose) The transpose, A T , of a matrix A is the matrix obtained from A by writing its rows as columns. If A is an k×n matrix and B = A T then B is the n×k matrix with b ij = a ji . If A T =A, then A is symmetric 7

9. Example: 8

10. DETERMINANT OF MATRIX Determinant is a scalar Defined for a square matrix Is the sum of selected products of the elements of the matrix, each product being multiplied by +1 or -1 9

11. Inverse of a matrix Definition: An inverse matrix A -1 which can be found only for a square and a non-singular matrix A ,is a unique matrix satisfying the relationship AA -1 = I =A -1 A The formula for deriving the inverse is 10

12. Calculation of Inversion using Determinants Example: find the inverse of the matrix Solve: 11

13. Systems of Equations in Matrix Form The system of linear equations: can be rewritten as the matrix equation Ax=b, where If an n×n matrix A is invertible, then it is nonsingular, and the unique solution to the system of linear equations Ax=b is x=A -1 b. 12

14. Example: solve the linear system 13

15. Unique properties of matrices In normal algebra , if we multiply two non-zero values, then the outcome will never be a zero . But if we multiply two non-zero values in matrix , then the outcome can be zero. 14

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17. Field of Geology ● Taking seismic surveys ● Plotting graphs & statistics ● Scientific analysis Application of Matrix In our everyday life 16

18. Field of Statistics & Economics ● Presenting real world data such as People's habit, traits & survey data ● Calculating GDP Field of Animation ● Operating 3D software & Tools ● Performing 3D scaling/Transforming ● Giving reflection, rotation 17

19. ANY QUESTIONS?