[MATRICES ]

Matrix (plural: matrices) a rectangular array of numbers written within brackets Represented with a capital letter and classify by its dimension Dimensions of a Matrix/Order of a Matrix determine by the number of horizontal rows and the number of vertical columns Matrix Element each number in a matrix

Writing the dimensions of a matrix. 4 columns A= 3 rows Matrix A is a 3 x 4 matrix.

Write the dimensions or order of each matrix. 1. 3. 2. 4. 6 5 -3 -7 1 0 9 -4 1/3 -3 3 X 3 1 X 3 10 0 1 -5 -6.2 9 5 0 -2 0.5 17 3 X 2 2 X 3

Identifying a Matrix Element aij denotes the element of the matrix A on the ith row and jth column. Example: Identify element a13 in Matrix A. Answer: a13 means the element in row 1, column 3. a13 = 1

Identify each matrix element

Adding and Subtracting Matrices to add or subtract matrices A and B with the same dimensions, add or subtract the corresponding elements ***Note: you can only add or subtract matrices with the same dimensions.

Properties: Matrix Addition If A, B, and C are m x n matrices, then a. Closure Property A + B is an m x n matrix Commutative Property A + B = B + A Associative Property for Addition (A + B) + C = A + (B + C) Additive Identity Property There exist a unique m x n matrix O such that O + A = A + O = A Additive Inverse Property For each A, there exists a unique opposite –A. A + (-A) = O

Find the sum or difference of each matrix. 1. 1. -2 0 3 -5 7 9 -3 -9 6 12 + 4 7 -3 -6 1 19 = 2. -12 24 -3 5 -1 10 -3 1 2 -4 -1 5 + -15 25 -1 1 -2 15 = 3. -9 7 -2 1 8 -4 3 0 6 5 10 - -12 7 -8 -4 -2 = 4. -3 5 -1 10 -3 1 2 -4 - 0 4 -3 14 =

Identify whether the two matrices are additive inverse or not. 1. 5 0 -2 -14 -5 0 2 , Yes. Find the additive inverse of the given matrix. 1. -1 10 -5 0 2 -3 -10 5 0 -2 3 =>

Solving Matrix Equations an equation in which the variable is a matrix Equal Matrices matrices with the same dimensions and with equal corresponding elements

Solving a Matrix Equation Solve for the matrix X. Solution: 1 3 2 0 1 8 9 = X - 1 3 2 0 1 8 9 = X - 1 3 2 0 1 8 9 + X = 1 2 11 11 X =

Do these… = - X = Solve for Matrix X. 1. X + 2. -1 0 2 5 10 7 -4 4 7 -1 0 2 5 10 7 -4 4 = X + 7 -6 -1 Answer: X = 2 1 -1 0 2 1 11 3 -13 15 -9 8 - X = -9 -2 12 -15 -11 -7 Answer: X =

Determining Equal Matrices Determine whether the two matrices in each pair are equal. 1. 2. 4 6 8 , 8/2 18/3 16/2 No, because they do not have the same dimensions. -2 3 5 0 -8/4 6 – 3 15/3 4 - 4 , Yes, because they have the same dimensions and the corresponding elements are equal.

Finding Unknown Matrix Elements Solve the equation for x and y. x + 8 -5 3 -y 38 -5 3 4y – 10 = Solution: x + 8 = 38 x = 30 -y = 4y – 10 -5y = -10 y = 2

Do these… , = Solve each unknown variable in each equation. 1. 2. 3x 4 -9 x + y , x = -3, y = 7 4 8 12 4x – 6 -10t + 5x 4x 15t +1.5x = x = 2; t = 3/5