4-2 Adding & Subtracting Matrices Objective: To add and subtract matrices

Adding Matrices The table shows information on ticket sales for a new movie that is showing at two theaters. Sales are for children (C) and adults (A). Theater C A C A 1 198 350 54 439 2 201 375 58 386 Matinee Evening a. Write two 2  2 matrices to represent matinee and evening sales. Theater 1 198 350 Theater 2 201 375 Matinee C A Theater 1 54 439 Theater 2 58 386 Evening C A

Continued (continued) b. Find the combined sales for the two showings. 198 350 201 375 + 54 439 58 386 = 198 + 54 350 + 439 201 + 58 375 + 386 = Theater 1 252 789 Theater 2 259 761 C A

Finding Identity and Inverse Matrices Find each sum. 9 0 –4 6 0 0 3 –8 –5 1 –3 8 5 –1 a. + b. + = 9 + 0 0 + 0 –4 + 0 6 + 0 = 3 + (–3) –8 + 8 –5 + 5 1 + (–1) = 9 0 –4 6 = 0 0

Subtracting Matrices 4 8 –2 0 7 –9 4 5 A = and B = . Find A – B. 4 8 –2 0 7 –9 4 5 A = and B = . Find A – B. Method 1: Use additive inverses. A – B = A + (–B) = + 4 8 –2 0 –7 9 –4 –5 Write the additive inverses of the elements of the second matrix. 4 + (–7) 8 + 9 –2 + (–4) 0 + (–5) Add corresponding elements = –3 17 –6 –5 = Simplify.

Continued (continued) Method 2: Use subtraction. A – B = – 4 8 –2 0 4 8 –2 0 7 –9 4 5 4 – 7 8 – (–9) –2 – 4 0 – 5 Subtract corresponding elements = –3 17 –6 –5 = Simplify.

Solving Matrix Equations 2 5 3 –1 8 0 10 –3 –4 9 6 –9 Solve X – = for the matrix X. X – = 10 –3 –4 9 6 –9 2 5 3 –1 8 0 X – + = + 2 5 3 –1 8 0 10 –3 –4 9 6 –9 Add to each side of the equation. 12 2 –1 8 14 –9 X = Simplify.

Determining Equal Matrices Determine whether the matrices in each pair are equal. a. M = ; N = 8 + 9 5 –6 –1 0 0.7 17 5 4 – 10 –2 + 1 0 – 79 M = ; N = 8 + 9 5 –6 –1 0 0.7 17 5 4 – 10 –2 + 1 0 – 7 9 Both M and N have three rows and two columns, but – 0.7. M and N are not equal matrices. 7 9 = /

Continued (continued) 27 9 – 16 4 3 –4 40 –3 b. P = ; Q = 8 0.2 12 4 3 –4 40 –3 b. P = ; Q = 8 0.2 12 4 P = ; Q = 3 –4 40 –3 – 27 9 16 4 12 8 0.2 Both P and Q have two rows and two columns, and their corresponding elements are equal. P and Q are equal matrices.

Finding Unknown Matrix Elements Solve the equation 2m – n –3 8 –4m + 2n = for m and n. 15 m + n 8 –30 2m – n = 15 –3 = m + n –4m + 2n = –30 2m – n –3 8 –4m + 2n = 15 m + n 8 –30 Since the two matrices are equal, their corresponding elements are equal.

Continued (continued) Solve for m and n. 2m – n = 15 m + n = –3 3m = 12 Add the equations. m = 4 Solve for m. 4 + n = –3 Substitute 4 for m. n = –7 Solve for n. The solutions are m = 4 and n = –7.

Homework Pg 178 # 2,3,4,10,11,14,17