Lesson 11: Matrix Addition Is Commutative

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M2 PRECALCULUS AND ADVANCED TOPICS

Classwork Opening Exercise Kiamba thinks 퐴 + 퐵 = 퐵 + 퐴 for all 2 × 2 matrices. Rachel thinks it is not always true. Who is correct? Explain.

Exercises 1. In two-dimensional space, let 퐴 be the matrix representing a rotation about the origin through an angle of 45°, and 1 let 퐵 be the matrix representing a reflection about the 푥-axis. Let 푥 be the point ( ). 1 a. Write down the matrices 퐴, 퐵, and 퐴 + 퐵.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M2 PRECALCULUS AND ADVANCED TOPICS

b. Write down the image points of 퐴푥, 퐵푥, and (퐴 + 퐵)푥, and plot them on graph paper.

c. What do you notice about (퐴 + 퐵)푥 compared to 퐴푥 and 퐵푥?

2. For three matrices of equal size, 퐴, 퐵, and 퐶, does it follow that 퐴 + (퐵 + 퐶) = (퐴 + 퐵) + 퐶? a. Determine if the statement is true geometrically. Let 퐴 be the matrix representing a reflection across the 푦-axis. Let 퐵 be the matrix representing a counterclockwise rotation of 30°. Let 퐶 be the matrix representing 1 a reflection about the 푥-axis. Let 푥 be the point ( ). 1

b. Confirm your results algebraically.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M2 PRECALCULUS AND ADVANCED TOPICS

c. What do your results say about matrix addition?

푥 0 3. If 푥 = (푦), what are the coordinates of a point 푦 with the property 푥 + 푦 if the origin 푂 = (0)? 푧 0

11 −5 2 4. Suppose 퐴 = (−34 6 19) and matrix 퐵 has the property that 퐴푥 + 퐵푥 is the origin. What is the matrix 퐵? 8 −542 0

5. For three matrices of equal size, 퐴, 퐵, and 퐶, where 퐴 represents a reflection across the line 푦 = 푥, 퐵 represents a 1 counterclockwise rotation of 45°, 퐶 represents a reflection across the 푦-axis, and 푥 = ( ): 2 a. Show that matrix addition is commutative: 퐴푥 + 퐵푥 = 퐵푥 + 퐴푥

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M2 PRECALCULUS AND ADVANCED TOPICS

b. Show that matrix addition is associative: 퐴푥 + (퐵푥 + 퐶푥) = (퐴푥 + 퐵푥) + 퐶푥

6. Let 퐴, 퐵, 퐶, and 퐷 be matrices of the same dimensions. Use the commutative property of addition of two matrices to prove 퐴 + 퐵 + 퐶 = 퐶 + 퐵 + 퐴.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M2 PRECALCULUS AND ADVANCED TOPICS

Problem Set

1. Let 퐴 be a matrix transformation representing a rotation of 45° about the origin and 퐵 be a reflection across the 푦-axis. Let 푥 = (3,4). a. Represent 퐴 and 퐵 as matrices, and find 퐴 + 퐵. b. Represent 퐴푥 and 퐵푥 as matrices, and find (퐴 + 퐵)푥. c. Graph your answer to part (b). d. Draw the parallelogram containing 퐴푥, 퐵푥, and (퐴 + 퐵)푥.

2. Let 퐴 be a matrix transformation representing a rotation of 300° about the origin and 퐵 be a reflection across the 푥-axis. Let 푥 = (2, −5). a. Represent 퐴 and 퐵 as matrices, and find 퐴 + 퐵. b. Represent 퐴푥 and 퐵푥 as matrices, and find (퐴 + 퐵)푥. c. Graph your answer to part (b). d. Draw the parallelogram containing 퐴푥, 퐵푥, and (퐴 + 퐵)푥.

3. Let 퐴, 퐵, 퐶, and 퐷 be matrices of the same dimensions. a. Use the associative property of addition for three matrices to prove (퐴 + 퐵) + (퐶 + 퐷) = 퐴 + (퐵 + 퐶) + 퐷. b. Make an argument for the associative and commutative properties of addition of matrices to be true for finitely many matrices being added.

th th 4. Let 퐴 be a 푚 × 푛 matrix with element in the 푖 row, 푗 column 푎푖푗, and 퐵 be a 푚 × 푛 matrix with element in the th th 푖 row, 푗 column 푏푖푗. Present an argument that 퐴 + 퐵 = 퐵 + 퐴.

5. For integers 푥, 푦, define 푥 ⊕ 푦 = 푥 ⋅ 푦 + 1; read “푥 plus 푦” where 푥 ⋅ 푦 is defined normally. a. Is this form of addition commutative? Explain why or why not. b. Is this form of addition associative? Explain why or why not.

6. For integers 푥, 푦, define 푥 ⊕ 푦 = 푥. a. Is this form of addition commutative? Explain why or why not. b. Is this form of addition associative? Explain why or why not.

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