Properties of Matrix Operations

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

Let $A,B,C$ be matrices of appropriate sizes, $O$ be the zero matrix, and $r,s$ be scalars.

◇Details / Theory

Let $A,B,C$ be matrices of appropriate sizes, $O$ be the zero matrix, and $r,s$ be scalars.

◇Matrix Addition Properties:

1. Commutative: $A+B=B+A$ (Handout 1, Page 13)
2. Associative: $(A+B)+C=A+(B+C)$ (Handout 1, Page 15)
3. Additive Identity: $A+O=A$ (Handout 1, Page 16)
4. Additive Inverse: For each $A$, there exists a unique matrix $-A$ such that $A+(-A)=O$. (Handout 1, Page 17)

◇Scalar Multiplication Properties:

1. Associative: $(rs)A=r(sA)$ (Handout 1, Page 18)
2. Distributive: $r(A+B)=rA+rB$ and $(r+s)A=rA+sA$ (Handout 1, Page 19)
3. Scalar Identity/Zero: $1A=A$, $0A=O$, $rO=O$ (Handout 1, Page 20)

◇Matrix Multiplication Properties:

1. Associative: $(AB)C=A(BC)$ (Provided sizes are compatible) (Handout 1, Page 23)
   PN: Associativity means we can group multiplications, but we cannot change the order.

2. Distributive: $A(B+C)=AB+AC$ (Left) and $(A+B)C=AC+BC$ (Right) (Provided sizes are compatible) (Handout 1, Page 24)

3. Scalar Interaction: $(rA)B=r(AB)=A(rB)$ (Handout 1, Page 26)

◇Transpose of a Matrix Properties:

1. Transpose of Sum: $(A+B)^T=A^T+B^T$ (Handout 1, Page 29)

2. Transpose of Product: $(AB)^T=B^T A^T$ (Note the reversal of order!) (Handout 1, Page 29)
   PN: This reversal of order for the transpose of a product is extremely important and often tested.

3. Transpose of Transpose: $(A^T)^T=A$ (Handout 1, Page 29)

4. Transpose of Scalar Multiple: $(rA)^T=rA^T$ (Handout 1, Page 29)

◇Common Pitfalls

Matrix multiplication is NOT commutative: $AB \neq BA$ in general.
$(A + B)^2 \neq A^2 + 2AB + B^2$ unless $AB = BA$.
$(AB)^T = B^T A^T$ — note the reversed order.

◇Summary

TL;DR — Matrix operations are associative and distributive but multiplication is not commutative. Transpose reverses product order. These properties underpin all matrix algebra proofs.

◇Links

Related: - ◇Matrix Addition - ◇Scalar Multiplication - ◇Matrix Multiplication - ◇Transpose of a Matrix
Source / Reference:

◇References & Navigation

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• Related Notes: ◇Upper Triangular Matrix, ◇Identity Matrix, ◇Matrices Detailed