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

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

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 z ij = x ij + y ij, where x ij, y ij, z ij are the elements on the i-th row, j-th column of matrices X, Y and Z.

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.

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)

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. 2A2. –3B3. 3A – 10B

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

Matrix Product If A is an m × p matrix 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.

Check Sizes Before Multiplication

MATRIX PRODUCT

Example

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: C ij =

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

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: 1.Form the augmented matrix [A| I] 2.Perform row operations on [A| I] to get a matrix of the form [I | B]. 3.Matrix B is A -1.

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 -1 B. Example: 2x – 3y = 4 x + 5y = 2 Write matrices A, X, B in this example.

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

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

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) –1 D 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.