Matrices

Rectangular arrays of numbers used to represent linear transformations and systems of equations.

Definitions

A matrix is a rectangular array of real numbers enclosed by a pair of brackets, with $m$ rows and $n$ columns, denoted as $m \times n$.

$$A = [a_{ij}] = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$

where $a_{ij}$ refers to the element in the $i$-th row and $j$-th column.

Leading entry, $P_i$ — the first non-zero element from the left of the $i$-th row.

Leading diagonal — diagonal elements $a_{11}, a_{22}, \ldots, a_{mm}$ of the matrix.

Types of Matrices

Type	Definition	Notation / Example
Row matrix	A matrix with only one row	$(2 \quad 5 \quad 1)$
Column matrix	A matrix with only one column	$\begin{pmatrix} 1 \ 0 \ 6 \end{pmatrix}$
Square	Equal number of rows and columns ($m = n$)	$n \times n$
Zero	All elements are 0	$0$
Diagonal	Square matrix where all non-diagonal elements are 0	$\text{diag}(d_1, \ldots, d_n)$
Identity	Square matrix with 1s on principal diagonal and 0s elsewhere	$I_n$
Upper Triangular	Square matrix where all entries under the diagonal are 0
Lower Triangular	Square matrix where all entries above the diagonal are 0
Symmetric	Square matrix with $a_{ij} = a_{ji}$ for all $i, j$; i.e., $B^T = B$
Skew-symmetric	Square matrix where $B^T = -B$ and $b_{ii} = 0$

Matrix Types and Operations Mindmap

mindmap
  root((Matrices))
    Types
      Row Matrix
      Column Matrix
      Square Matrix
      Zero Matrix
      Diagonal Matrix
      Identity Matrix
      Triangular Matrix
        Upper Triangular
        Lower Triangular
      Symmetric Matrix
      Skew-Symmetric Matrix
    Operations
      Addition and Subtraction
      Scalar Multiplication
      Matrix Multiplication
      Transpose
    Key Concepts
      Determinant
      Inverse
      Elementary Row Operations
      Solving Linear Systems

Matrix Operations

Addition and Subtraction

Element-wise for matrices of same dimension: $$(A \pm B){ij} = a{ij} \pm b_{ij}$$

Scalar Multiplication

The product of a scalar $k$ and a matrix $A$, written $kA$, is the matrix obtained by multiplying each element of $A$ by $k$.

$$(kA){ij} = k \cdot a{ij}$$

Properties:

• $k(A + B) = kA + kB$
• $(k_1 + k_2)A = k_1A + k_2A$
• $k_1(k_2A) = k_2(k_1A) = (k_1k_2)A$

Matrix Multiplication

Multiplication between two matrices $A$ and $B$, $AB$, can only be done if the number of columns of $A$ equals the number of rows of $B$.

If $A$ is of order $m \times p$ and $B$ is of order $p \times n$, then $AB$ is of order $m \times n$:

$$(AB){ij} = \sum{k=1}^{p} a_{ik} \cdot b_{kj}$$

The principle 'row into column' is used to obtain each element of the product.

Properties:

• NOT commutative: $AB \neq BA$ (in general)
• Distributive: $A(B + C) = AB + AC$
• If $A$ is a zero matrix of order $m \times n$, $B$ is of order $n \times p$, then $AB = 0$
• Identity: $AI = IA = A$
• Powers: $A^m = A \cdot A \cdot \ldots \cdot A$ ($m$ times), for square matrix $A$
• Law of exponents: $A^p A^q = A^{p+q}$, $(A^p)^q = A^{pq}$ for $p > 0, q > 0$
• Identity powers: $I = I^2 = I^3 = \cdots = I^n$

Transpose

Let $A$ be an $m \times n$ matrix, the transpose of $A$ written as $A^T$, is an $n \times m$ matrix obtained by interchanging the rows and columns of $A$.

$$(A^T){ij} = a{ji}$$

Properties:

• $(kA)^T = kA^T$, $k$ a scalar
• $(A^T)^T = A$
• $(A \pm B)^T = A^T \pm B^T$
• $(AB)^T = B^T A^T$

Determinant

Notation: $|A|$ or $\det(A)$

2×2 Matrix

Let $A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}$ then:

$$|A| = \begin{vmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}$$

Minor and Cofactor

If $A$ is a square matrix of order $3 \times 3$, the minor of $a_{ij}$, denoted by $M_{ij}$, is the determinant of the $2 \times 2$ matrix obtained by deleting the $i$-th row and $j$-th column.

The cofactor of $a_{ij}$ is denoted by $C_{ij}$ and:

$$C_{ij} = (-1)^{i+j} M_{ij}$$

Note: For a $3 \times 3$ matrix, the sign of the cofactors are: $$\begin{pmatrix} + & - & + \ - & + & - \ + & - & + \end{pmatrix}$$

3×3 Matrix

Diagonal Expansion (for checking)

For checking purposes, the determinant of $3 \times 3$ matrix $A$ can be evaluated by diagonal expansion:

$$|A| = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}$$

Cofactor Expansion

The determinant of a $3 \times 3$ matrix $A$ is the product of $a_{ij}$ and $C_{ij}$ of one of the rows or columns of $A$.

Based on $i$-th row: $$|A| = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} = \sum_{j=1}^{3} a_{ij}C_{ij}$$

Based on $j$-th column: $$|A| = a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} = \sum_{i=1}^{3} a_{ij}C_{ij}$$

Properties of Determinants

1. If $A$ is an $n \times n$ matrix and $k$ is a scalar, then $|kA| = k^n |A|$
2. If $A$ and $B$ are two square matrices, then $|AB| = |A||B|$
3. $|A| = |A^T|$ (determinant unchanged by transpose)
4. If two rows or columns are interchanged, the sign of the determinant is changed
5. The value of the determinant is unchanged by interchanging rows and columns
6. If any two rows or columns are identical, then the value of the determinant is zero
7. If $A$ is a triangular matrix, then $|A|$ is the product of the elements on the leading diagonal

Singularity:

• If $|A| = 0$, $A$ is singular (no inverse exists)
• If $|A| \neq 0$, $A$ is non-singular (inverse exists)

Matrix Inverse

For square matrix $A$, inverse $A^{-1}$ satisfies: $$AA^{-1} = A^{-1}A = I$$

Inverse exists iff: $|A| \neq 0$ (non-singular). If $|A| = 0$, $A$ is singular and has no inverse.

2×2 Inverse

$$A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \quad \text{where } |A| = ad - bc$$

Shortcut for 2×2:

1. Interchange the elements of the leading diagonal ($a \leftrightarrow d$)
2. Reverse the sign of the other elements ($b \rightarrow -b$, $c \rightarrow -c$)
3. Divide by the determinant

General Formula (Adjoint Method)

$$A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)$$

where $\text{adj}(A) = C^T$ is the adjoint (transpose of the cofactor matrix)

Inverse via Elementary Row Operations (ERO)

Write the augmented matrix $(A|I)$ and apply ERO until it becomes $(I|A^{-1})$.

Key property: $(AB)^{-1} = B^{-1}A^{-1}$

Elementary Row Operations (ERO)

There are three elementary row operations:

1. Interchange any two rows: $R_i \leftrightarrow R_j$
2. Multiply all elements of a row by a scalar: $R_i \rightarrow kR_i$
3. Multiply a row by a scalar and add to another row: $R_j \rightarrow kR_i + R_j$

When $A$ is changed to $B$ using ERO, the matrices are equivalent.

ERO are used to:

• Find matrix inverses: $(A|I) \rightarrow (I|A^{-1})$
• Solve linear systems: $(A|B) \rightarrow (I|X)$ (Gauss-Jordan)

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Solving Linear Systems

Matrix form: $AX = B$

• $A$: coefficients matrix
• $X$: variables matrix
• $B$: constants matrix

Method 1: Inverse Matrix

$$X = A^{-1}B$$

Cannot be used if $A$ is singular ($|A| = 0$).

Method 2: Gauss-Jordan Elimination

1. Write the system as $AX = B$
2. Form the augmented matrix $(A|B)$
3. Use ERO to reduce to $(I|X)$

Method 3: Cramer's Rule

Uses determinants. For $n$ equations and $n$ variables: $x_i = \frac{|A_i|}{|A|}$ where $A_i$ is $A$ with column $i$ replaced by $B$.

See Cramer's Rule for detailed theory, worked examples (2×2, 3×3), word problems, and comparison with other methods.

Solution Types

1. Unique solution: $|A| \neq 0$ (system is consistent and independent)
2. Infinitely many solutions: $|A| = 0$ and $(\text{adj } A)B = 0$ (consistent, dependent)
3. No solution: $|A| = 0$ and $(\text{adj } A)B \neq 0$ (inconsistent)

Gaussian Elimination Flowchart

graph TD
    Start([Start]) --> WriteSystem["Write system as AX = B"]
    WriteSystem --> Augmented["Form augmented matrix (A|B)"]
    Augmented --> ERO["Apply ERO to obtain row echelon form"]
    ERO --> Consistent{"Consistent system?"}
    Consistent -->|"No"| NoSolution["No Solution"]
    Consistent -->|"Yes"| Pivots{"Pivots =<br/>variables?"}
    Pivots -->|"Yes"| BackSub["Back Substitution"]
    BackSub --> Unique["Unique Solution"]
    Pivots -->|"No"| FreeVars["Express in terms of<br/>free variables"]
    FreeVars --> Infinite["Infinitely Many Solutions"]

    style Start fill:#e7f5ff,stroke:#1971c2
    style Consistent fill:#ffe8cc,stroke:#d9480f
    style Pivots fill:#ffe8cc,stroke:#d9480f
    style NoSolution fill:#ffe3e3,stroke:#c92a2a
    style Unique fill:#d3f9d8,stroke:#2f9e44
    style Infinite fill:#fff4e6,stroke:#e67700

Related Sources

• FAD1015 L27-L28 — Matrices (Types, Operations & Determinants)
• FAD1015 L29-L30 — Matrices (Inverse & Systems of Equations)

Related Courses

• FAD1015 - Mathematics III