Matrices Detailed

Tags: #◇Problem Solving #◇Matrices #◇Linear Algebra #◇CCDSTRU #◇M7

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◇Important Formulas & Concepts

Matrix: A rectangular array of numbers. A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix.
Square Matrix: An $n \times n$ matrix.
Notation: Boldface uppercase letters (e.g., $\mathbf{A}$) are used to represent matrices, while their elements are represented with lowercase letters (e.g., $a_{ij}$ for the element at row $i$, column $j$).

Matrix Addition

Let $\mathbf{A}$ and $\mathbf{B}$ be $m \times n$ matrices. The sum $\mathbf{A} + \mathbf{B}$ is the $m \times n$ matrix calculated by adding corresponding elements. Formula: $\mathbf{A} + \mathbf{B} = [a_{ij} + b_{ij}]$

[!warning] Shape Requirement
Matrices must have the exact same dimensions to be added together.

Matrix Multiplication

Let $\mathbf{A}$ be an $m \times k$ matrix and $\mathbf{B}$ be a $k \times n$ matrix. The product $\mathbf{AB}$ is an $m \times n$ matrix $\mathbf{C} = [c_{ij}]$. Formula: $c_{ij} = \sum_{x=1}^{k} a_{ix}b_{xj}$ In simpler terms: The element at row $i$ and column $j$ of the product is the dot product of the $i$-th row of $\mathbf{A}$ and the $j$-th column of $\mathbf{B}$.

[!warning] Shape Requirement
To multiply $\mathbf{A} \times \mathbf{B}$, the number of columns in $\mathbf{A}$ must equal the number of rows in $\mathbf{B}$. Furthermore, matrix multiplication is generally not commutative ($\mathbf{AB} \neq \mathbf{BA}$).

◇Worked Examples

Example 1: Matrix Addition
Given:
$\mathbf{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix}$

$\mathbf{A} + \mathbf{B} = \begin{bmatrix} 1+2 & 2+4 & 3+6 \\ 4+1 & 5+3 & 6+5 \end{bmatrix} = \begin{bmatrix} 3 & 6 & 9 \\ 5 & 8 & 11 \end{bmatrix}$

Example 2: Matrix Multiplication
Given a $2 \times 3$ matrix $\mathbf{A}$ and a $3 \times 2$ matrix $\mathbf{B}$:
$\mathbf{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 2 & 4 \\ 3 & 6 \\ 4 & 8 \end{bmatrix}$
The product $\mathbf{AB}$ will be a $2 \times 2$ matrix:
$\mathbf{AB} = \begin{bmatrix} (1\times2 + 2\times3 + 3\times4) & (1\times4 + 2\times6 + 3\times8) \\ (4\times2 + 5\times3 + 6\times4) & (4\times4 + 5\times6 + 6\times8) \end{bmatrix}$
$= \begin{bmatrix} 2+6+12 & 4+12+24 \\ 8+15+24 & 16+30+48 \end{bmatrix} = \begin{bmatrix} 20 & 40 \\ 47 & 94 \end{bmatrix}$

◇References & Navigation

• Next Topic: ◇Mathematical Induction
• Related Notes: ◇Matrices and Matrix Operations, ◇Definition of a Matrix