12.1 Addition of Matrices

Matrix: is any rectangular array of numbers written within brackets; represented by a capital letter; classified by its dimensions Dimensions are the rows × columns Ex: 3 × 24 × 1 column matrix 2 × 2 square matrix 1 × 4 row matrix Each number in a matrix is called an element. We use subscripts to identify position in the matrix, a ij Ex: in matrix A, a 32 is: –7

Two matrices are equal iff they have the same dimensions and all of their corresponding elements are equal Matrix Addition If two matrices, A and B, have the same dimensions, then their sum A + B is a matrix of the same dimensions whose elements are the sums of the corresponding elements of A and B. Properties of Matrix Addition If A, B and C are m × n matrices, then A + B is an m × n matrixClosure A + B = B + ACommutative (A + B) + C = A + (B + C)Associative There exists a unique m × n matrix O such that O + A = A + O = A Additive Identity For each A, there exists a unique matrix, –A, such that A + –A = O Additive Inverse Matrix Subtraction If two matrices, A and B, have the same dimensions, then A – B = A + (–B). *Basically match up elements & add

Ex 1) a) Find A + B b) Find A – B= A + (–B) On Your Own c) Find B – A = B + (–A)

We can multiply a scalar times a matrix. Properties of Scalar Multipication If A, B and O are m × n matrices and c and d are scalars, then cA is an m × n matrix Closure (cd)A = c(dA) Associative 1·A = A Multiplicative Identity 0A = O and cO = O Multiplicative Property of the zero scalar and the zero matrix c(A + B) = cA + cB (c + d)A = cA + dA Distributive Properties

Matrices can be used to solve many real world problems. Ex 2) Carl & Flo are training for a triathlon by running, cycling & swimming. The matrices below show the number of miles that each devotes to each activity, both on weekdays & weekend days. What is the total number of miles that each devotes to each activity in a 7-day week? Weekday Carl Flo Running A = Cycling Swimming Running B = Cycling Swimming Weekend Carl Flo Carl: 52 mi running, 330 mi cycling & 24 mi swimming Flo: 66 mi running, 290 mi cycling & 16 mi swimming

You can also solve a “matrix equation.” 2 ways  (1) thinking algebraically & treating matrix as a whole (2) Element by Element (we will do both ways) Ex 3) Solve Method 1: First distribute the 3 Method 2: 2x + 3 = 1 2x = –2 x = –1 2x + 6 = –2 2x = –8 x = –4 2x + 12 = 6 2x = –6 x = –3 2x + 3 = 5 2x = 2 x = 1

Homework #1201 Pg 600 #4, 5, 8, 10, 14, 16, 19, 20, 23, 25, 27, 29, 35, 37, 39, 43