Chapter 6 Systems of Linear Equations and Matrices Sections 6.3 – 6.5

1. Chapter 6 Systems of Linear Equations and MatricesSections 6.3 – 6.5

2. 6.3 Basic Matrix Operations • Size of a matrix • Row matrix • Column matrix • Square matrix • Element of matrix A: aij : element in row i and column j

3. Sum of two matrices • Sum of two matrices of the same size: Given matrices X and Y (both have the same size m  n). Matrix Z = X + Y has elements zij = xij + yij, where xij , yij, zij are the elements on the i-th row, j-th column of matrices X, Y and Z.

4. Additive inverse of a matrix A is the matrix –A in which each element is the additive inverse of the corresponding element of A. • Zero matrix O: all elements are zeros. • Identity property: A + O = O + A = O, A is any matrix.

5. Subtraction: The difference of X and Y (same size) is matrix Z, in which each element is the difference of the corresponding elements of X and Y, or, equivalently: Z = X – Y = X + (– Y)

6. Product of a scalar k and a matrix X is the matrix kX, each of whose elements is k times the corresponding element of X. Exercise: • Let Find each of the following: 1. 2A 2. –3B 3. 3A – 10B

7. Product of a Row Matrix and a Column Matrix 6.4 MATRIX PRODUCT AND INVERSE

8. Matrix Product If A is an m ×pmatrix and B is a p×n matrix, then the matrix product of A and B, denoted AB, is an m ×n matrix whose element in the ith row and jth column is the real number obtained from the product of the ith row of A and the jth column of B. If the number of columns in A does not equal the number of rows in B, then the matrix product AB is not defined.

9. Check Sizes Before Multiplication

10. MATRIX PRODUCT 7-1-67

11. Example

12. Product (Sigma Notation) • Let A be an mn matrix and let B be an nk matrix. The product matrix AB (denoted C) is the mk matrix whose entry in the i-th row and j-th column is: Cij =

13. Properties • Associative property: A(BC) = (AB)C, A+(B+C) = (A+B)+C • Distributive property: A(B+C) = AB + AC • Identity matrix I: On the main diagonal: all elements are 1 Elsewhere: all elements are 0 • Not commutative: AB  BA in general

14. Definition of inverse matrix: • Given matrix A, if exists matrix B so that AB = I, B is called inverse matrix, and denoted A-1 (read A-inverse). • Singular, non-singular matrix Inverse matrix calculation: • Form the augmented matrix [A| I] • Perform row operations on [A| I] to get a matrix of the form [I | B]. • Matrix B is A-1.

15. 6.5 Applications of Matrices 1. Solving systems with matrices: System AX = B, where A is coefficient matrix, X is the matrix of variables, and B is the matrix of constants, is solved by first finding A-1. Then, if A-1 exists, X = A-1B. Example: 2x – 3y = 4 x + 5y = 2 Write matrices A, X, B in this example.

16. 6.5 Applications of Matrices 2. Input-output analysis • Input-output matrix A (or technological matrix) of an economy.Example 3.

17. 6.5 Applications of Matrices 2. Input-output analysis • Production matrix X • Demand matrix D = X – AX Example 4.

18. 6.5 Applications of Matrices 2. Input-output analysis • In practice, A and D are known, we need to find the production matrix: X–1 = (I – A) –1D Example 6: An economy depends on 2 basic products: wheat and oil. To produce 1 ton of wheat requires .25 ton of wheat and .33 ton of oil. The production of 1 ton of oil consumes .08 ton of wheat and .11 ton of oil. Find the production that will satisfy the demand of 500 ton of wheat and 1000 ton of oil.