Algebraic Properties of Matrix Operations
- 1. Algebraic Properties of Matrix Operations
- 2. Determine what property of real numbers is being described in the following: 1. 𝑎 + 𝑏 = 𝑏 + 𝑎 a. Commutative Property of Addition b. Associative Property of Addition c. Additive Inverse d. Identity Property of Addition Answer: a. Commutative Property of Addition. The order of the numbers does not affect the sum.
- 3. Determine what property of real numbers is being described in the following: 2. 𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑎 a. Commutative Property of Addition b. Associative Property of Addition c. Additive Inverse d. Identity Property of Addition Answer: b. Associative Property of Addition. The way the numbers are paired does not affect the sum.
- 4. Determine what property of real numbers is being described in the following: 3. 𝑎 + 0 = 𝑎 a. Commutative Property of Addition b. Associative Property of Addition c. Additive Inverse d. Identity Property of Addition Answer: d. Identity Property of Addition. When zero is added to any number, the number remains unchanged.
- 5. Determine what property of real numbers is being described in the following: 4. 𝑎 + (−𝑎) = 0 a. Commutative Property of Addition b. Associative Property of Addition c. Additive Inverse / Inverse Property of Addition d. Identity Property of Addition Answer: c. Additive Inverse / Inverse Property of Addition. The sum of the number and its additive inverse (opposite) is zero.
- 6. Determine what property of real numbers is being described in the following: 5. 𝑎(𝑏𝑐) = 𝑎𝑏 𝑐 a. Commutative Property of Multiplication b. Associative Property of Multiplication c. Multiplicative Inverse / Inverse Property of Multiplication d. Identity Property of Multiplication Answer: b. Associative Property of Multiplication. The way the numbers are paired does not affect the product.
- 7. Determine what property of real numbers is being described in the following: 7. 𝑐 𝑎 + 𝑏 = 𝑐𝑎 + 𝑐𝑏 a. Distributive Property b. Associative Property of Multiplication c. Multiplicative Inverse / Inverse Property of Multiplication d. Identity Property of Multiplication Answer: a. Distributive Property. Multiplication can be distributed over addition or subtraction.
- 8. Matrices Some of the properties of matrix operations are similar to properties for real numbers.
- 11. Properties of Matrix Addition
- 12. Theorem 1.1. Properties of Matrix Addition Let 𝐴, 𝐵, and 𝐶 𝑏𝑒 𝑚 × 𝑛 matrices. 𝑎 𝐴 + 𝐵 = 𝐵 + 𝐴 (Commutative Property of Matrix Addition) 𝑏 𝐴 + 𝐵 + 𝐶 = 𝐴 + 𝐵 + 𝐶. (Associative Property of Matrix Addition) 𝑐 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑢𝑛𝑖𝑞𝑢𝑒 𝑚 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑂 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴 + 0 = 𝐴 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑚 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴, 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑢𝑛𝑖𝑞𝑢𝑒 𝑚 × 𝑛 𝑧𝑒𝑟𝑜 𝑚𝑎𝑡𝑟𝑖𝑥. (Identity Property of Matrix Addition) 𝑑 𝐹𝑜𝑟 𝑒𝑎𝑐ℎ 𝑚 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴, 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑢𝑛𝑖𝑞𝑢𝑒 𝑚 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝐷 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴 + 𝐷 = 0. 𝑊𝑒 𝑠ℎ𝑎𝑙𝑙 𝑤𝑟𝑖𝑡𝑒 𝐷 𝑎𝑠 − 𝐴, 𝑠𝑜 A + D = 0 can be written as 𝐴 + −𝐴 = 0. 𝑇ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 − 𝐴 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑡ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝐴. 𝑊𝑒 𝑎𝑙𝑠𝑜 𝑛𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 − 𝐴 𝑖𝑠 −1 𝐴.
- 13. Examples:
- 14. Examples:
- 15. Properties of Matrix Multiplication
- 17. Theorem 1.2. Properties of Matrix Multiplication 𝑎 𝐼𝑓 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑝𝑝𝑟𝑜𝑝𝑟𝑖𝑎𝑡𝑒 𝑠𝑖𝑧𝑒𝑠, 𝑡ℎ𝑒𝑛 A 𝐵𝐶 = 𝐴𝐵 𝐶. 𝑏 𝐼𝑓 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑝𝑝𝑟𝑜𝑝𝑟𝑖𝑎𝑡𝑒 𝑠𝑖𝑧𝑒𝑠, 𝑡ℎ𝑒𝑛 𝐴 + 𝐵 𝐶 = 𝐴𝐶 + 𝐵𝐶. 𝑐 𝐼𝑓 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑝𝑝𝑟𝑜𝑝𝑟𝑖𝑎𝑡𝑒 𝑠𝑖𝑧𝑒𝑠, 𝑡ℎ𝑒𝑛 𝐶 𝐴 + 𝐵 = 𝐶𝐴 + 𝐶𝐵.
- 21. Properties of Scalar Multiplication
- 23. Theorem 1.3. Properties of Scalar Multiplication If 𝑟 and 𝑠 are real numbers and 𝐴 and 𝐵 are matrices of the appropiate sizes, then 𝑎 𝑟 𝑠𝐴 = 𝑟𝑠 𝐴 𝑏 𝑟 + 𝑠 𝐴 = 𝑟𝐴 + 𝑠𝐴 𝑐 𝑟 𝐴 + 𝐵 = 𝑟𝐴 + 𝑟𝐵 𝑑 𝐴 𝑟𝐵 = 𝑟 𝐴𝐵 = 𝑟𝐴 𝐵.
- 27. Theorem 1.4. Properties of Transpose If 𝑟 𝑖𝑠 𝑎 𝑠𝑐𝑎𝑙𝑎𝑟 and 𝐴 and 𝐵 are matrices of the appropiate sizes, then 𝑎 𝐴𝑇 𝑇 = 𝐴 𝑏 𝐴 + 𝐵 𝑇 = 𝐴𝑇 + 𝐵𝑇 𝑐 𝐴𝐵 𝑇 = 𝐵𝑇𝐴𝑇 𝑑 𝑟𝐴 𝑇 = 𝑟𝐴𝑇.
- 32. 1.Introductory Linear Algebra with Applications 7th edition By: Bernard Kolman and David R. Hill (2002)