Chapter 2Chapter 2Systems of Linear Equations Systems of Linear Equations

and Matricesand Matrices

Section 2.4Section 2.4

Multiplication of MatricesMultiplication of Matrices

Writing Systems of Equations in Writing Systems of Equations in Abbreviated FormAbbreviated Form

Consider the following system of equations Consider the following system of equations with three unknowns.with three unknowns.

2x + y – z = 22x + y – z = 2 x + 3y + 2z = 1x + 3y + 2z = 1 x + y + z = 2x + y + z = 2

This system can be written in an abbreviated This system can be written in an abbreviated form asform as

What is a Matrix?What is a Matrix? A A matrixmatrix is a rectangular array of numbers is a rectangular array of numbers

enclosed by brackets.enclosed by brackets. Each number in the array is an Each number in the array is an elementelement or or

entryentry.. An An augmented matrixaugmented matrix separates the separates the

constants in the last column of the matrix constants in the last column of the matrix from the coefficients of the variables with from the coefficients of the variables with a vertical line.a vertical line.

Classifications of MatricesClassifications of Matrices

Often named with capital letters.Often named with capital letters. Classified by size (the number of Classified by size (the number of

rows and columns they contain).rows and columns they contain). A matrix with A matrix with mm rows and rows and nn columns columns

is an is an mm x x nn matrix. The number of matrix. The number of rows is always given first.rows is always given first.

Special Types of MatricesSpecial Types of Matrices

A matrix with the same number of A matrix with the same number of rows as columns is called a rows as columns is called a square square matrixmatrix..

A matrix containing only one row is A matrix containing only one row is called a called a row matrixrow matrix or a or a row vectorrow vector..

A matrix of only one column is a A matrix of only one column is a column matrixcolumn matrix or a or a column vectorcolumn vector..

Scalar MultiplicationScalar Multiplication When determining the product of a real When determining the product of a real

number and a matrix, the real number is number and a matrix, the real number is called a called a scalarscalar..

ExampleExample

Find the product of each of the following.Find the product of each of the following.

1.) -5A1.) -5A 2.) 2B2.) 2B

5 7

4 2A

4 2 2

6 3 8

1 5 12

B

Matrix MultiplicationMatrix Multiplication

ExampleExample

SolutionSolution

AnswerAnswer

Notice that when you multiply a 2 X 3 matrix with a 3 X 1 matrix, the product is a 2 X 1 matrix.

Another ExampleAnother Example

SolutionSolution

AnswerAnswer

CAUTION!!!CAUTION!!! Sometimes the product of two Sometimes the product of two

matrices matrices does not existdoes not exist!!

The product The product ABAB of two matrices of two matrices AA and and BB can be found only if the can be found only if the number of columns of number of columns of AA is the same is the same as the number of rows of as the number of rows of BB..

The final product will have as many The final product will have as many rows as rows as AA and as many columns of and as many columns of BB..

Examples for Us!Examples for Us!

Use the matrices defined above to find the following products, if they exist.

1.) AF 2.) AC 3.) DE

4.) ED 5.) BD 6.) EA