Operations on Matrices

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

◇Overview

Matrix operations define how matrices can be combined or transformed. The fundamental operations are addition, subtraction, scalar multiplication, matrix multiplication, and transposition.

◇Key Concepts

Addition/Subtraction: Requires matrices of the same size Scalar multiplication: Multiply every entry by a constant Matrix multiplication: $A_{m \times n} \cdot B_{n \times p} = C_{m \times p}$ (inner dimensions must match) Transpose: Swap rows and columns

◇Details / Theory

Summary of Operations

Operation	Requirement	Result Size
◇Matrix Addition $A + B$	Same size $m \times n$	$m \times n$
◇Matrix Subtraction $A - B$	Same size $m \times n$	$m \times n$
◇Scalar Multiplication $kA$	Any matrix	Same as $A$
◇Matrix Multiplication $AB$	Cols of $A$ = Rows of $B$	$m \times p$
◇Transpose $A^T$	Any matrix	$n \times m$

Key Properties

Addition is commutative: $A + B = B + A$ Multiplication is not commutative: $AB \neq BA$ in general Multiplication is associative: $(AB)C = A(BC)$ Distributive: $A(B + C) = AB + AC$ $(AB)^T = B^T A^T$ (reverse order!)

Important Warnings

$AB = AC$ does not imply $B = C$ (no cancellation law in general)
$AB = O$ does not imply $A = O$ or $B = O$
$(A + B)^2 \neq A^2 + 2AB + B^2$ unless $AB = BA$

◇Common Pitfalls

Attempting to add matrices of different sizes. Assuming matrix multiplication is commutative. Forgetting the transpose reverses order of products: $(AB)^T = B^T A^T$ .

◇Summary

TL;DR — The five core matrix operations are addition, subtraction, scalar multiplication, matrix multiplication, and transpose. Addition is commutative; multiplication is not. Watch dimension requirements.

◇Links

Previous Topic: ◇Nonsingular Matrix
Next Topic: ◇Polynomial in a Matrix
Related Notes: ◇Polynomial in a Matrix, ◇Elementary Row Operation, ◇Matrices Detailed

OPERATIONS ON MATRICES