1. Chapter 2 Systems of Linear Equations and Matrices Section 2.4 Multiplication of Matrices

2. Writing Systems of Equations in Abbreviated Form Consider the following system of equations with three unknowns. Consider the following system of equations with three unknowns. 2x + y – z = 2 x + 3y + 2z = 1 x + 3y + 2z = 1 x + y + z = 2 x + y + z = 2 This system can be written in an abbreviated form as

3. What is a Matrix? A matrix is a rectangular array of numbers enclosed by brackets. A matrix is a rectangular array of numbers enclosed by brackets. Each number in the array is an element or entry. Each number in the array is an element or entry. An augmented matrix separates the constants in the last column of the matrix from the coefficients of the variables with a vertical line. An augmented matrix separates the constants in the last column of the matrix from the coefficients of the variables with a vertical line.

4. Classifications of Matrices Often named with capital letters. Often named with capital letters. Classified by size (the number of rows and columns they contain). Classified by size (the number of rows and columns they contain). A matrix with m rows and n columns is an m x n matrix. The number of rows is always given first. A matrix with m rows and n columns is an m x n matrix. The number of rows is always given first.

5. Special Types of Matrices A matrix with the same number of rows as columns is called a square matrix. A matrix with the same number of rows as columns is called a square matrix. A matrix containing only one row is called a row matrix or a row vector. A matrix containing only one row is called a row matrix or a row vector. A matrix of only one column is a column matrix or a column vector. A matrix of only one column is a column matrix or a column vector.

6. Scalar Multiplication When determining the product of a real number and a matrix, the real number is called a scalar. When determining the product of a real number and a matrix, the real number is called a scalar.

7. Example Find the product of each of the following. Find the product of each of the following. 1.) -5A2.) 2B

8. Matrix Multiplication

9. Example

10. Solution

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

12. Another Example

13. Solution

14. Answer

15. CAUTION!!! Sometimes the product of two matrices does not exist! The product AB of two matrices A and B can be found only if the number of columns of A is the same as the number of rows of B. The final product will have as many rows as A and as many columns of B.

16. Examples for Us! Use the matrices defined above to find the following products, if they exist. 1.) AF2.) AC3.) DE 4.) ED5.) BD6.) EA