This document summarizes key algebraic properties of matrix operations that are analogous to properties of real numbers. It describes the commutative, associative, identity, and inverse properties of matrix addition and multiplication. It also covers the distributive, scalar, and transpose properties as they relate to matrix operations. Examples are provided to illustrate the properties. The document serves to establish the foundational algebraic structure of matrices and their 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.

25. Properties of Transpose

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)