Matrices: Definitions and Operations

Matrices Overview

• Matrices involve adding, subtracting, multiplying, and dividing numbers.
• Prerequisites include elementary row operations.
• They are used in solving systems of equations and describing geometric transformations.
• Used to derive addition formulae and for balancing chemical equations.
• They help analyze flight stopover data and model electrical circuits.
• Utilized for formulating fundamental physical laws.

Basic Matrix Definitions

• Matrices are rectangular arrays of numbers called entries or elements.
• Entries are organized in rows and columns within parentheses.
• Matrix order is defined as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.
• $A = (a_{ij}){m \times n}$ represents a matrix where $a{ij}$ is the element in the $i$-th row and $j$-th column.
• $a_{ij}$ denotes the element in row $i$ and column $j$, referred to as the $(i, j)^{th}$ entry of matrix A.
• Course primarily focuses on $2 \times 2$ and $3 \times 3$ matrices.

Special Matrix Types

• The main diagonal consists of entries $a_{ij}$ where $i = j$.
• Special cases arise when $m = 1$ (row matrix) or $n = 1$ (column matrix).
• A row matrix is a $1 \times n$ matrix written as $(a_{11} \ a_{12} \ ... \ a_{1(n-1)} \ a_{1n})$.
• A column matrix is an $m \times 1$ matrix.
• A column matrix is written as $\begin{pmatrix} a_{11} \ a_{21} \ \vdots \ a_{(m-1)1} \ a_{m1} \end{pmatrix}$.
• Square matrices ($m = n$) have equal rows and columns.
• A square matrix is written as
  $\begin{pmatrix} a_{11} & a_{12} & \dots & a_{1m} \ a_{21} & a_{22} & \dots & a_{2m} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \dots & a_{mm} \end{pmatrix}$.
• Identity matrices ($I_m$) are $m \times m$ matrices with 1s on the main diagonal and 0s elsewhere.
• An identity matrix can be written as $\begin{pmatrix} 1 & 0 & \dots & 0 \ 0 & 1 & \dots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & & 1 \end{pmatrix}$.
• Zero matrices ($O_{m \times n}$) are $m \times n$ matrices with all entries equal to 0.
• A zero matrix can be written as $\begin{pmatrix} 0 & 0 & 0 & \dots & 0 \ 0 & 0 & 0 & \dots & 0 \ 0 & 0 & 0 & \dots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & & 0 \end{pmatrix}$.
• $O_m$ denotes a square zero matrix.

Matrix Algebra: Addition, Subtraction, and Scalar Multiplication

• Matrix sum of A and B involves adding corresponding entries: $(a + b){ij} = a{ij} + b_{ij}$.
• Matrix difference involves subtracting corresponding entries: $(a - b){ij} = a{ij} - b_{ij}$.
• Matrix addition and subtraction are only possible if A and B have the same order.
• Scalar multiplication of A by k multiplies each entry of A by k: $(ka){ij} = ka{ij}$, where $k \in \mathbb{R}$.

Matrix Multiplication Definition

• The matrix product of $A$ and $B$, where $A$ is of order $m \times n$ and $B$ is of order $n \times p$, is obtained as $(ab){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$.
• The number of columns in A must equal the number of rows in B.
• To compute the matrix product AB, split A into rows and B into columns
• The $(i, j)^{th}$ entry of AB, obtained from the 'scalar product' of the $i^{th}$ row of A with the $j^{th}$ column of B.
• In general, $AB \ne BA$
• In the context of forming $AB$, matrix $B$ is pre-multiplied by matrix $A$, while $A$ is post-multiplied by $B$

Matrix Transpose

• The transpose of $A$ (order $m \times n$), denoted as $A^T$, is an $n \times m$ matrix formed by interchanging rows and columns of $A$.
• $(a^T){ij} = a{ji}$

Matrix Power

• A square matrix $A$ can be multiplied by itself any number of times.
• $A^n = \underbrace{A \times A \times A \times \dots \times A}_{n \text{ times}}$

Basic Matrix Properties

• Commutative Property of Addition: $A + B = B + A$
• Associative Property of Addition: $(A + B) + C = A + (B + C)$
• Distributive Properties: $k(A + B) = kA + kB$
• Transpose of a sum: $(A + B)^T = A^T + B^T$
• Transpose of a transpose: $(A^T)^T = A$
• Transpose of a scalar multiple: $(kA)^T = kA^T$
• Associative Property of Multiplication: $A(BC) = (AB)C$
• Distributive Property: $A(B + C) = AB + AC$
• Transpose of a product: $(AB)^T = B^T A^T$
• Product of exponential matrices: $A^m A^n = A^{m+n} = A^n A^m$
• Additive Identity Property: $A + O = O + A = A$
• Multiplicative Identity Property: $AI = IA = A$
• Multiplication by Zero Matrix: $AO = AO = O$
• Identity and zero matrices behave like the numbers 1 and 0 in ordinary arithmetic and algebra.

Special Types of Matrices: Symmetric

• A matrix $A$ is symmetric if $A^T = A$
• Symmetric matrices must be square.

Special Types of Matrices: Skew-Symmetric

• A matrix $A$ is skew-symmetric (aka anti-symmetric) if $A^T = -A$
• Skew-symmetric matrices must be square with all diagonal entries equal to 0.
• The sum of 2 skew-symmetric matrices is skew-symmetric.
• Proof: If $A$ and $B$ are skew-symmetric, $(A+B)^T = A^T + B^T = -A - B = -(A+B)$

Orthogonal Matrices

• A square matrix $A$ (of order $n \times n$) is orthogonal if $A^T A = I_n$, where $I_n$ is the identity matrix of order $n$.

Determinants

• For a $1 \times 1$ system $ax = b$, the solution is $x = \frac{b}{a}$, assuming $a \ne 0$.
• For a $2 \times 2$ system $\begin{cases} ax + by = e \ cx + dy = f \end{cases}$, the solutions are $x = \frac{de - bf}{ad - bc}$ and $y = \frac{af - ce}{ad - bc}$, assuming $ad - bc \ne 0$.
• For a $3 \times 3$ system: $\begin{cases} ax + by + cz = j\ dx + ey + fz = k \ gx + hy + iz = l \end{cases}$, the solutions are $x = \frac{j(ei-fh) - k(bi - ch) + l(bf - ce)}{a(ei - fh) - b(di - fg) + c(dh - eg)}$, $y = \frac{j(fg-di) - k(cg - ai) + l(cd - af)}{a(ei - fh) - b(di - fg) + c(dh - eg)}$, $z = \frac{j(dh - ge) - k(ah - bg) + l(ae - bd)}{a(ei - fh) - b(di - fg) + c(dh - eg)}$ assuming $a(ei - fh) - b(di - fg) + c(dh - eg) \ne 0$.
• A permutation, denoted by $\sigma$, of an ordered set of numbers ($1, 2, 3, \dots, n$) is a rearrangement of those numbers.
• An even permutation involves an even number of consecutive switches.
• An odd permutation involves an odd number of consecutive switches.
• The sign of a permutation $\sigma$ ($sign \ \sigma$) is defined to be +1 for an even permutation and -1 for an odd permutation.
• The determinant of an $n \times n$ matrix is given by the Leibniz formula.
• $det(A) = |A| = \mathop{\sum}\limits_{\sigma} (sign \ \sigma) \prod_{i=1}^n a_{i,\sigma(i)}$
• The minor of entry $a_{ij}$, denoted as $M_{ij}$, is the determinant of the $(n-1) \times (n-1)$ matrix formed by deleting the $i^{th}$ row and $j^{th}$ column of $A$
• The cofactor of entry $a_{ij}$ is the quantity $C_{ij} = (-1)^{i+j} M_{ij}$
• These formulas can be written using cofactors via the Laplace expansion theorem.
• The determinant of an $n \times n$ matrix is given by the Laplace expansion formula
• $det(A) = \sum_{j=1}^{n} a_{ij} C_{ij}$ for $(i = 1, 2, 3, \dots, n)$

Determinant Theorems

• The determinant of a $1 \times 1$ matrix $A \equiv (a)$ is $|A| = a$
• The determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ is $|A| = ad - bc$
• The determinant of a $3 \times 3$ matrix $A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}$ is $|A| = a \begin{pmatrix} e & f \ h & i \end{pmatrix} - b \begin{pmatrix} d & f \ g & i \end{pmatrix} + c \begin{pmatrix} d & e \ g & h \end{pmatrix}$

Determinant Properties

• If $A$ and $B$ are $n \times n$ matrices and $k \in \mathbb{R}$:
  • $det(AB) = det(A)det(B)$
  • $det(kA) = k^n det(A)$
  • $det(A^T) = det(A)$

Inverse of a Matrix

• An $n \times n$ matrix $A$ has an inverse if there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I_n$.
• If $A$ has an inverse, $A$ is invertible (non-singular).
• If $A$ does not have an inverse, $A$ is non-invertible (singular).
• A matrix has only one inverse.
• The cofactor matrix of a square matrix A is the matrix $C$ such that its $(i,j)^{th}$ entry is $C_{ij}$
• The adjugate (aka classical adjoint) of a square matrix $A$ is given by $adj(A) = C^T$
• The inverse of a matrix A is $A^{-1} = \frac{adj(A)}{det(A)}$
• A matrix is invertible if and only if $det(A) \ne 0$

Properties of Inverse Matrices

• $(AB)^{-1} = B^{-1}A^{-1}$
• $(A^{-1})^T = (A^T)^{-1}$
• $det(A^{-1}) = \frac{1}{det(A)}$
• $(kA)^{-1} = \frac{1}{k}A^{-1}$

Transformation Matrices

• A geometrical transformation changes points in space.
• This course focuses on transformations in the xy-plane.
• A linear transformation in the plane maps a point P(x, y) to Q(ax + by, cx + dy) where (a, b, c, d ∈ ℝ).
• Matrix Representation: $\begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$ is a transformation matrix.
• The matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$ is called the transformation matrix.

Standard Transformations

• Reflection in the x-axis: $\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$
• Reflection in the y-axis: $\begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix}$
• Reflection in the line $y = x$: $\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$
• Anticlockwise rotation about the origin by angle θ: $\begin{pmatrix} cos \ \theta & -sin \ \theta \ sin \ \theta & cos \ \theta \end{pmatrix}$
• Dilatation (scaling) with factor k: $\begin{pmatrix} k & 0 \ 0 & k \end{pmatrix}$, where $k \in \mathbb{R}$
• If $k > 1$ the dilatation is an enlargement. If $k < 1$ then the dilatation is a reduction. If $k<0$, then the dilatation inverts

Combining Transformations

• Transformation matrices are usually combined to yield the resultant transformation.

Invariant Points

• An invariant point is a point whose coordinates stay the same after a transformation: $T\bold{x} = \bold{x}$

General equations

• If an $n \times n$ system of equations $A \bold{x} = \bold{b}$, then $\bold{x} = A^{-1} \bold{b}$
• A system of n equations in n unknowns has a solution if $det A \ne0 $