2.3 Modeling Real World Data with Matrices Objectives: Model data using matrices. Add, subtract, and multiply matrices.

A rectangular array of terms called elements. In a matrix they are m x n A matrix with m rows and n columns. “m by n” A matrix with only one row. A matrix with only one column. A matrix with the same number of rows as columns. Matrix: Dimensions: m x n matrix: Row matrix: Column matrix: Square matrix:

a23 is the element in the second row and third column. a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44

Ex. 1 During the summer of 2000, Ms. Robbins received several types of grains on her farm to feed her livestock. a.) Use a matrix to represent the data. June – 15,000 bushels of corn - 2,000 bushels of soybeans - 500 bushels of oats July – 13,500 bushels of corn - 6,500 bushels of soybeans - 1,000 bushels of oats Aug. - 14,000 bushels of corn - 5,500 bushels of soybeans - 1500 bushels of oats. b) Use a symbol to represent the number of bushels of soybeans in August.

Equal Matrices: Two matrices are equal iff they have the same dimensions and are identical, element by element. Ex. 2) Find the values of x and y for which the matrix equation is: y 4x y – 3 2x + 1 The sum of two m x n matrices is an m x n matrix in which the elements are the sum of the corresponding elements of the given matrices. Addition of Matrices:

Subtraction of Matrices: Ex. 3) Find A + B if A= -7 4 and B = 6 10 5 0 8 -9 3 -1 -2 5 The difference A – B of two m x n matrices is equal to the sum of A + (-B), where –B represents the additive inverse of B. Subtraction of Matrices: Find S - T if S = 2 -1 3 and T = -5 -4 1 -4 -2 -8 7 -8 4 Ex. 4)

Scalar: The number multiplied by a matrix. Scalar Product: The product of a scalar k and an m x n matrix A is an m x n matrix denoted by kA. Each element of kA equals k times the corresponding element of A. Ex. 5) If A = 5 -2 Find 4A 3 8 -1 -9

Product of Two Matrices: The product of an m x n matrix A and an n x r matrix B is an m x r matrix AB. The ijth element of AB is the sum of the products of the corresponding elements in the ith row of A and the jth column of B. *Can only multiply if the number of columns in A is the same as the number of rows in B. Product of Two Matrices:

Use matrices A = 4 -1 2 0 1 0 3 -2 4 B = 4 2 C = 1 2 -3 -2 3 3 1 0 AB Ex. 6.) Use matrices A = 4 -1 2 0 1 0 3 -2 4 B = 4 2 C = 1 2 -3 -2 3 3 1 0 AB b.) BC