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Welcome. To. A Session On. Mathematics for Decision Making. Your Text Book. BY Mathematics for Decision Making. Chapter 10. What is the role of Linear Algebra?. Linear Algebra permits expression of a complicated system of equations in Succinet, simplified way,

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  1. Welcome To A Session On Mathematics for Decision Making

  2. Your Text Book BY Mathematics for Decision Making Chapter 10

  3. What is the role of Linear Algebra? • Linear Algebra permits expression of a complicated system of equations in Succinet, simplified way, • It provides a shorthand method to determine whether a solution exists before it is attempted • It furnishes the means of solving the equation system. • Linear Algebra can be applied only to systems of linear equations.

  4. What is Matrix? A matrix is a rectangular array of numbers, parameters, or variables, each of which has a carefully ordered place within the Matrix. What are elements, rows and columns? The numbers (parameters or variables) contained in the matrix are called elements. The numbers in the horizontal line are called rows, and the numbers in the vertical line a are called columns.

  5. What are meant by square matrix, column and row vector? If the number of rows equals the number of columns (i.e.r=c), the matrix is called square matrix. If the matrix is composed of a single column (i.e. r×i), it is a column vector. If the matrix is composed of a single row (i.e.1×c), it is row vector.

  6. Examples Square Matrix: Column vector: Row vector:

  7. Addition and subtraction of matrices? • In the case of addition, each element of one matrix is added to corresponding element of another matrix. • In the case of subtraction, each element of one matrix is subtracted from corresponding element of another matrix. • In these cases, the dimensions of the matrices will have to be equal

  8. Examples N.B .If the matices are not of equal dimensions, they are not for addition or subtractions.

  9. When is Multiplication of Matrices comfortable? When Number of columns is the first the matrix (lead mation) is equal to the number of rows in the second matrix (i.e. e, =r2), two matrices will be comfortable for multiplication.

  10. How can two matrices be multiplied? Each row rector in the lead matrix is multiplied by each corresponding vector of the log matrix and then summed the products. 1strow/1st matrix ×1st column/2nd matrix =1st column element of the product 2ndrow /1st matrix ×1st column/2nd matrix =2nd column element of the product Similarly 1strow/1st matrix ×1st column/2nd matrix =1st column element of the product 2nd row /1st matrix ×1st column/2nd matrix =2ndcolumn element of the product

  11. How is the dimension of the resulting product matrix pre-determined? 2 × 3 = 3 ×2 2 Decision No. of columns will be equal to no. of rows (2×2) in the product matrix.

  12. Problem Are A and C conformable for multiplication? Solution: The number of columns in the first matrix is 3. The number of rows in the second matrix is 2. In other words 2×3≠2×3 they are not equal. Hence A and C are not conformable Thus, AC is not defined. Continued……

  13. What is a null matrix? If all the elements of a matrix are Os, the matrix is called a null matrix What is an identity matrix? An identity matrix is a square matrix which has 1 for every element on the principal diagonal form left to right and 0 everywhere else.

  14. What is the main property of an identity matrix? Multiplication of an identity matrix by itself leaves the identity matrix unchanged.

  15. What is transpose matrix? A matrix which converts the rows of A to columns and the columns of A to rows is called the transpose of A and is designated by

  16. What is a symmetric matrix? Any matrix for which A = A’ is a symmetric matrix A symmetric matrix for which A×A =A is an idempotent matrix. The identity matrix is symmetric and idempotent.

  17. How can the system of linear equations be expressed in matrix form? The equations 8x1+9x2=70 3x1+7x2=33 Can be expressed in matrix form Ax =B, where Here A= coefficient matrix X=Solution rector B=Vector of constant terms Continued……

  18. Problem: Determine Ak, given Here K is a scaler, and scaler multiplication is possible with a matrix of any dimension Hence the product is defined. Continued…………

  19. What is a singular matrix? A singular matrix is one in which a row or a column is a multiple of another row or column. Examples Continued…………

  20. Problem: A clothing store discounts all its slacks, jackets and suits by 20 per cent at the end of the year. If V1 is the value of stock in its three branches price to discount, find the value V2 after the discount, when Continued……..

  21. A 20 per cent discount means selling price of clothing at the end of the year is 80 per cent of its original value.

  22. Problem A company sells 700 CDs , 400 cassets, and 200 CD players each week. The selling price of CDx is $ 4, cassettes $6 and CD players $ 150. The cost to the shop is $ 3.25 for a CD, $4.75 for a casette, and $ 125 for a CD player. Find weekly profits by using by total and per unit concepts. a) But this is not defined i.e. not conformable, because the number of columns ≠ the number of rows taking the transpose of either P or Q will make the vectors conformable for multiplication.

  23. Continued……..

  24. b)

  25. Thank you For Attending The Session

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