9.2 Using Properties of Matrices Matrix- a rectangular arrangement of numbers in rows and columns Named with capital letters [ A ] Dimensions- row x column (2 x 4)

Element- each number in the matrix Named with lowercase letters and subscripts (a₂₃) First number of subscript= row Second number of subscript= column

Using Matrices to Represent Points/Figures in Coordinate Planes Use 2 row matrices to represent the vertices on a 2-dimensional figure 1st row = x-coordinate 2nd row= y-coordinate Example: Point (2,3)= 2 3

Represent figures using matrices Write a matrix to represent the point or polygon. a. Point A b. Quadrilateral ABCD

Represent figures using matrices SOLUTION a. Point matrix for A x-coordinate y-coordinate –4 b. Polygon matrix for ABCD x-coordinates y-coordinates –4 –1 4 3 0 2 1 –1 A B C D

Adding and subtracting matrices 5 –3 6 –6 1 2 3 –4 + a. 5 + 1 –3 + 2 6 + 3 –6 + (– 4) = 6 –1 9 –10 = 6 8 5 4 9 –1 – 1 –7 0 4 –2 3 b. 6 – 1 8 –(–7) 5 – 0 4 – 4 9 – (– 2) –1 – 3 = 5 15 5 0 11 –4 =

Examples: Adding and subtracting matrices In Exercises 3 and 4, add or subtract. 3. [–3 7] + [2 –5] SOLUTION [–3 7] + [2 –5] =[–1 2 ] 4. 1 –4 3 –5 – 2 3 7 8 SOLUTION 1 –4 3 –5 – 2 3 7 8 –1 –7 –4 –13 =

Represent a translation using matrices 1 5 3 The matrix represents ∆ABC. Find the image 1 0 –1 matrix that represents the translation of ∆ABC 1 unit left and 3 units up. Then graph ∆ABC and its image. SOLUTION The translation matrix is –1 –1 –1 3 3 3

Represent a translation using matrices Add this to the polygon matrix for the preimage to find the image matrix. + 1 5 3 1 0 –1 Polygon matrix A B C 0 4 2 4 3 2 = Image matrix A′ B′ C′ –1 –1 –1 3 3 3 Translation matrix

Multiplying matrices equal Number of columns in matrix A= Number of rows in column B A X B = A B (mxn) x (nxp) (m x p) equal 1 0 4 5 2 –3 –1 8 . Multiply The matrices are both 2 X 2, so their product is defined. Use the following steps to find the elements of the product matrix. SOLUTION

Multiplying matrices STEP 1 Multiply the numbers in the first row of the first matrix by the numbers in the first column of the second matrix. Put the result in the first row, first column of the product matrix. 1 0 4 5 2 –3 –1 8 1(2) + 0(–1) ? = ? ?

Multiplying matrices STEP 2 Multiply the numbers in the first row of the first matrix by the numbers in the second column of the second matrix. Put the result in the first row, second column of the product matrix. 1 0 4 5 2 –3 –1 8 1(2) + 0(–1) 1(–3) + 0(8) = ? ?

Multiplying matrices STEP 3 Multiply the numbers in the second row of the first matrix by the numbers in the first column of the second matrix. Put the result in the second row, first column of the product matrix. 1 0 4 5 –1 8 2 –3 1(2) + 0(–1) 1(–3) + 0(8) = 4(2) + 5(–1) ?

Multiplying matrices STEP 4 Multiply the numbers in the second row of the first matrix by the numbers in the second column of the second matrix. Put the result in the second row, second column of the product matrix. 1 0 4 5 2 –3 –1 8 = 4(2) + 5(–1) 4(–3) + 5(8) 1(2) + 0(–1) 1(–3) + 0(8)

Multiplying matrices STEP 5 Simplify the product matrix. 4(2) + 5(–1) 4(–3) + 5(8) 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 3 28 =

Multiplying Matrices Use the matrices below. Is the product defined? Explain. –3 4 A = C = 6.7 0 –9.3 5.2 B = [2 1] 6. AB Yes; the number of columns in A is equal to the number of rows in B. ANSWER

Multiplying Matrices Use the matrices below. Is the product defined? Explain. –3 4 A = C = 6.7 0 –9.3 5.2 B = [2 1] 1. BA ANSWER Yes; the number of columns in B is equal to the number of rows in A.

Multiplying Matrices Use the matrices below. Is the product defined? Explain. –3 4 A = C = 6.7 0 –9.3 5.2 B = [2 1] 2. AC ANSWER No; the number of columns in A is not equal to the number of rows in C.

Examples of Multiplying Matrices 1 0 3 8 0 1 –4 7 SOLUTION = 3 8 –4 7 1 0 0 1 3 8 –4 7 = 0(3) + 1(–4) 0(8) + 1(7) 1(3) + 0(–4) 1(8) + 0(7)

Solve a real-world problem SOFTBALL Two softball teams submit equipment lists for the season. A bat costs $20, a ball costs $5, and a uniform costs $40. Use matrix multiplication to find the total cost of equipment for each team.

Solve a real-world problem SOLUTION First, write the equipment lists and the costs per item in matrix form. You will use matrix multiplication, so you need to set up the matrices so that the number of columns of the equipment matrix matches the number of rows of the cost per item matrix.

Solve a real-world problem EQUIPMENT COST TOTAL COST = Bats Balls Uniforms Dollars Women Men Uniforms 13 42 16 15 45 18 Bats Balls 20 5 40 ? =

Solve a real-world problem You can find the total cost of equipment for each team by multiplying the equipment matrix by the cost per item matrix. The equipment matrix is 2 3 and the cost per item matrix is 3 1, so their product is a 2 1 matrix. 13 42 16 15 45 18 20 5 40 13(20) + 42(5) + 16(40) = 15(20) + 45(5) + 18(40) = 1110 1245 The total cost of equipment for the women’s team is $1110, and the total cost for the men’s team is $1245. ANSWER