1. Chapter 11 Section 11.0 Review of Matrices

2. Matrices A matrix (despite the glamour of the movie) is a collection of numbers arranged in a rectangle or an array. We use variables like A, B, C, …, [capital letter] to stand for a matrix. We use what are called double scripted variables with a lower case letter of the matrix to refer to the entries in a matrix. The numbers in the subscript give the position the variable is located in with the first number referring to the row and the second number the column. The dimensions or order of the matrix is given in the form (number of rows)  (number of columns). Don't multiply leave it this way! We name this matrix A This matrix has 2 rows and 3 columns, or has order or dimension 2  3 "read 2 by 3". What do we name this? What is the entry b 21 ? What is the entry b 12 ? What is the variable for 4? What are the dimensions? B 9 7 b 32 3  3

3. Types of Matrices A matrix can have different adjectives that describe it depending on its dimensions. A matrix is called square matrix if it has the same number of rows and columns. A matrix is called row matrix if it has only one row (i.e. all of the entries are in a single row). A matrix is called a column matrix if it only has one column (i.e. all of the entries are in a single column). Give the dimensions of each matrix below and determine if it is a square, row or column matrix or if it does not fall in any of the categories. 4  2 None 3  3 Square 2  1 column 4  4 Square 1  1 Square row column 1  4 row 2  2 Square 2  4 None 1  2 row 2  5 None

4. Matrix Operations Adding & Subtracting Matrices The way that matrices are added or subtracted is to add or subtract their corresponding entries. This means that the matrices must be of the same dimensions or order. If they are not we say the two matrices are not the same dimensions we say the matrices are nonconformable. The matrices C and D are nonconformable. They can not be added even though they both have 6 entries. Matrix C is 2  3 Matrix D is 1  6

5. Multiplication by a Scalar We can multiply a matrix by a number (sometimes called a scalar) by multiplying each entry in the matrix by the number. This operation can always be done. We say it is always conformable. We can begin to combine more than one operation at a time. What you get here is nonconformable since the first matrix is 1  3 and the second matrix is 3  1.

6. Multiplying Matrices This is not as obvious an operation as you might think! It is not as easy as addition or subtraction that you get with the corresponding entries! What you do is to multiply each entry in a row on the matrix on the left with its corresponding entry in a column of the matrix on the right and add them up. AB = (rows of matrix A ) (columns of matrix B ) Look at the example below: The matrix A is 2  3 and the matrix B is 3  1. The number of columns for the matrix on the right must be the same as the number of rows for the matrix on the left or else they are nonconformable! 2  32  33  13  12  12  1 The dimensions of the result are given by the rows of A and columns of B.

7. 1  2 2  3 1  3 The matrix E and the matrix D are nonconformable even though they are the same dimensions. The columns and rows do not match up! 2  3 If you multiply a 2  2 matrix by a 2  1 matrix you get another 2  1 matrix!

8. Identity Matrices A matrix with the same number of rows and columns is called square. A square matrix with 1's down the top left to bottom right diagonal and 0's off that diagonal is called the identity matrix. They come in different size identity matrices. 2  2 3  3 4  4 An identity matrix has the property that if you multiply it either on the right or left by any conformable matrix you get the conformable matrix (i.e. I n A = A and AI n = A ). The matrix I n for matrices acts like the number 1 for numbers.

9. Representing Matrix Multiplication A movie theatre has two prices for movie admission, one for children and one for adult (people over 12). It also charges one rate for a matinee and another for an evening movie given in the table to the right. AdultChild Matinee$4$2 Evening$8$3 Find the cost of taking 2 adults and 4 children to a matinee movie. 2·4 + 4·2 = 8 + 8 =16 Find the cost of taking 2 adults and 4 children to an evening movie. 2·8 + 4·3 = 16 + 12 =28 A matrix is a rectangular array of numbers notice what we get if we multiply the two matrices below together. Cost of Matinee Movie Cost of Evening Movie In other words the matrix multiplication combines all of these calculations into one. This enables you to represent many different calculations at once.