MTH108 Business Math I Lecture 20

Chapter 9 Matrix Algebra

Objectives Provide an understanding of the nature of the matrix and matrix representation of data Provide an understanding of the algebra of matrices Provide applications of matrices and matrix algebra

Main Topics Introduction to matrices Special types of matrices Matrix operations The determinant The inverse of a matrix Some applications

Review Matrix: need; definition Basic facts: notation; elements; dimension; examples Types of matrices: row vector; column vector; square matrix; identity matrix; diagonal matrix; zero matrix Transpose of a matrix Matrix operations: addition; subtraction; scalar multiplication Properties of matrix operations

Today’s Topics Matrix operations; matrix multiplication Representation of system of equations

Matrix multiplication Assume that a matrix A having dimension is to be multiplied with matrix having dimension . The product C = AB is to be defined a matrix of order whose entry in ith row and jth column is obtained as: Sum the products formed by multiplying in order each entry in row I of A by the corresponding entry in column j of B.

Properties The matrix product AB is defined when rows of A is equal to the columns of B. The resulting product will have the order of row of A and column of B. The order of multiplication matters.

Examples 1) 2)

Examples 3)

Examples 3)

Examples 4) 5)

Examples 6)

Examples 7) Cost Vector Suppose that the prices for products A, B and C are given by the price vector If quantities of A, B and C that are purchased are given by the column vector

Examples 7) Cost Vector Then total cost C is given by:

Properties of matrix multiplication

Equality of matrices Definition: Two matrices A and B are equal if and only if, they have the same dimension and e.g.

A matrix equation can define a system of equations. E.g.

Solve the equation

Representation of an equation An equation may be represented using the matrix form. E.g.

Representation of an equation A linear equation of the form

Representation of system of equations A system of equations of the form

Consider the system of equations

The system of equations

Summary Matrix multiplication Properties Equality of matrices Matrix equation System of equations Section 9.3 Follow up exercises