“No one can be told what the matrix is… they have to see it for themselves…” - Lawrence Fishburne It’s a good thing we have Section 7.2a!!!

Let m and n be positive integers. An m x n matrix Definition: Matrix Let m and n be positive integers. An m x n matrix (read “m by n matrix”) is a rectangular array of m rows and n columns of real numbers. We also use the shorthand notation [a ] for this matrix. ij

Definition: Matrix Each element, or entry, a , of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a is in the i-th row and j-th column. In general the order of an m x n matrix is m x n. If m = n, the matrix is a square matrix. Two matrices are equal matrices if they have the same order and their corresponding elements are equal. ij ij

Practice Problem!!! Order 2 x 3 = –2 = 2 Determine the order of the matrix, and identify the specified elements. Order 2 x 3 = –2 = 2

Practice Problem!!! Order 4 x 2 = 4 Doesn’t exist! Determine the order of the matrix, and identify the specified elements. Order 4 x 2 = 4 Doesn’t exist!

Matrix Addition and Subtraction We can add or subtract matrices of the same order by adding or subtracting their corresponding entries. Matrices of different orders cannot be added or subtracted!!! Let and be matrices of order m x n. 1. The sum A + B is the m x n matrix 2. The difference A – B is the m x n matrix

Scalar Multiplication of Matrices When dealing with matrices, real numbers are scalars. The product of the real number k and the m x n matrix is the m x n matrix The matrix is a scalar multiple of A.

A Few More Definitions… Let A = [a ] be any m x n matrix. The m x n matrix O = [0] consisting entirely of zeros is the zero matrix because A + O = A. In other words, O is the additive identity for the set of all m x n matrices. The m x n matrix B = [–a ] consisting of the additive inverses of the entries of A is the additive inverse of A because A + B = O. ij ij

Practice Problem!!! For the given matrices, find (a) A + B, (b) A – B, (c) 3A, and (d) 2A – 3B.

Practice Problem!!! For the given matrices, find (a) A + B, (b) A – B, (c) 3A, and (d) 2A – 3B.

Practice Problem!!! Let A = [a ] and B = [b ] be 2 x 2 matrices with a = 3i – j and b = i + j – 3 for i = 1, 2, and j = 1, 2. ij ij ij 2 2 ij 1. Determine A and B.

Practice Problem!!! Let A = [a ] and B = [b ] be 2 x 2 matrices with a = 3i – j and b = i + j – 3 for i = 1, 2, and j = 1, 2. ij ij ij 2 2 ij 2. Determine the additive inverse –A of A and verify that A + (–A) = [0]. What is the order of [0]?  The order of [0] is 2 x 2.

Practice Problem!!! Let A = [a ] and B = [b ] be 2 x 2 matrices with a = 3i – j and b = i + j – 3 for i = 1, 2, and j = 1, 2. ij ij ij 2 2 ij 3. Determine 3A – 2B.

Matrix Multiplication

Definition: Matrix Multiplication Let A = [a ] be an m x r matrix and B = [b ] an r x n matrix. ij ij The product AB = [c ] is the m x n matrix where ij  To multiply two matrices, the columns of the first matrix must equal the rows of the second matrix. The resulting matrix has rows and columns determined by the “outside” values. Ex: Can we multiply a 3 x 2 matrix and a 2 x 4 matrix??? (3 x 2)(2 x 4) Yes, we can multiply… and the result is a 3 x 4 matrix…

Find the product AB, where possible:  Support with a calculator???

Find the product AB, where possible: The product AB is not defined!!! Why??

A florist makes three different cut flower arrangements (I, II, and III). Matrix A shows the number of each type of flower used in each arrangement. I II III Roses A = Carnations Lilies The florist can buy his flowers from two different wholesalers (W1 and W2), but wants to give all his business to one or the other. The cost of the three flower types from the two whole- salers is shown in matrix B. W1 W2 Roses B = Carnations Lilies

Construct a matrix showing the cost of making each of the three I II III W1 W2 Roses Roses A = Carnations B = Carnations Lilies Lilies Construct a matrix showing the cost of making each of the three flower arrangements from flowers supplied by the two different wholesalers. We want the columns of A to match up with the rows of B, so we first switch the rows and columns of A: Rose Carn Lily I The new matrix is called the transpose of A, and is denoted A T II III

Construct a matrix showing the cost of making each of the three I II III W1 W2 Roses Roses A = Carnations B = Carnations Lilies Lilies Construct a matrix showing the cost of making each of the three flower arrangements from flowers supplied by the two different wholesalers. T Now, we find the product A B: Rose Carn Lily W1 W2 W1 W2 I Rose I II x Carn = II III Lily III