FAD1015 L27-L28 — Matrices (Types, Operations & Determinants)

Lectures 27–28 introducing matrix algebra: types of matrices, basic operations, transpose, and determinants. Source file: (L27L28) - WEEK 16_MATRICES.pdf

Summary

Introduction to matrices: definitions, special types, fundamental operations (addition, subtraction, scalar multiplication, matrix multiplication), transpose, and determinants of $2 \times 2$ and $3 \times 3$ matrices including minors, cofactors, and determinant properties.

Key Concepts

Matrices — Matrix algebra fundamentals
Determinant — Scalar value associated with a square matrix

L27 — Matrices

1. Definition of a Matrix

A matrix is a rectangular array of real numbers enclosed by a pair of brackets.

Each matrix has its own size or order
The size is determined by the number of rows and columns
If a matrix $A$ has $m$ rows and $n$ columns, then $A$ is called an $m \times n$ matrix (read as "$m$ by $n$")

A general matrix of size $m \times n$ may be denoted by:

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

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.

2. 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 matrix	Equal number of rows and columns ($m = n$)	$n \times n$
Zero matrix	All elements are zero	$0$
Diagonal matrix	Square matrix where all non-diagonal elements are zero	$\text{diag}(d_1, \ldots, d_n)$
Identity matrix, $I_m$	Square matrix with 1s on the principal diagonal and 0s elsewhere	$I_n$
Upper triangular matrix	Square matrix where all entries under the diagonal elements are zero
Lower triangular matrix	Square matrix where all entries above the diagonal elements are zero
Symmetrical matrix	Square matrix with $a_{ij} = a_{ji}$ for all $i, j$; i.e., $B^T = B$
Skew-symmetrical matrix	Square matrix where $B^T = -B$ and $b_{ii} = 0$

mindmap
  root((Matrix Types))
    General Shape
      Row matrix
      Column matrix
      Square matrix
      Zero matrix
    Special Square Matrices
      Diagonal matrix
      Identity matrix I
      Upper triangular
      Lower triangular
      Symmetrical B^T = B
      Skew-symmetrical B^T = -B

3. Operations on Matrices

Addition and Subtraction

If $A = (a_{ij})$ and $B = (b_{ij})$ are two matrices of the same size, then $A + B$ is the resulting matrix with $(a + b){ij} = a{ij} + b_{ij}$. The difference $A - B$ is obtained by subtracting corresponding elements.

Properties:

$A + B = B + A$ → Commutative Property
$A + (B + C) = (A + B) + C$ → Associative Property
$A + 0 = 0 + A = A$
$A + (-A) = 0 = (-A) + A$

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

The product of a row matrix $(a \quad b)$ and a column matrix $\begin{pmatrix} c \ d \end{pmatrix}$ is defined by:

$$(a \quad b) \begin{pmatrix} c \ d \end{pmatrix} = ac + bd$$

For two matrices $A$ and $B$, the principle 'row into column' is used to obtain the product $AB$.

graph LR
    A["Matrix A<br/>order m x p"] --> CHECK{"Columns of A<br/>=<br/>Rows of B?"}
    B["Matrix B<br/>order p x n"] --> CHECK
    CHECK -->|Yes| PROD["Product AB<br/>order m x n"]
    CHECK -->|No| ERROR["Multiplication<br/>NOT defined"]

    style A fill:#e7f5ff,stroke:#1971c2
    style B fill:#e7f5ff,stroke:#1971c2
    style CHECK fill:#fff4e6,stroke:#e67700
    style PROD fill:#d3f9d8,stroke:#2f9e44
    style ERROR fill:#ffe3e3,stroke:#c92a2a

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

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

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

Properties of Multiplication:

$A(B + C) = AB + AC$
If $A$ is a zero matrix of order $m \times n$, $B$ is of order $n \times p$, then $AB = 0$
$AB \neq BA$ (matrix multiplication is not commutative)
If $A$ is a square matrix and $I$ is an identity matrix of the same order, then $AI = IA = A$
Let $A$ be a square matrix of order $n \times n$, then $A^2 = AA$. In general, $A^m = A \cdot A \cdot \ldots \cdot A$ ($m$ times)
The law of exponents is valid: $A^p A^q = A^{p+q}$, $(A^p)^q = A^{pq}$ for $p > 0, q > 0$
Let $I$ be an identity matrix, then $I = I^2 = I^3 = \cdots = I^n$

Transpose of a Matrix

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 of Transpose:

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

L28 — Determinant of Matrices

1. Determinant of a $2 \times 2$ Matrix

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

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

2. 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. Determinant of a $3 \times 3$ Matrix

Diagonal Expansion (for checking)

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

$$|A| = P_1 + P_2 + P_3 - P_4 - P_5 - P_6$$ $$= a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}$$

Cofactor Expansion

The determinant of a $3 \times 3$ matrix $A$ is the product of $a_{ij}$ and $C_{ij}$ of one of the row or column 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:

$$|A| = a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} = \sum_{i=1}^{3} a_{ij}C_{ij}$$

4. Properties of Determinants

If $A$ is an $n \times n$ matrix and $k$ is a scalar, then $|kA| = k^n |A|$
If $A$ and $B$ are two square matrices, then $|AB| = |A||B|$
The determinants of matrix $A$ and its transpose $A^T$ are equal: $|A| = |A^T|$
If two rows or columns are interchanged, the sign of the determinant is changed
The value of the determinant is unchanged by interchanging rows and columns
If any two rows or columns are identical, then the value of the determinant is zero
If $A$ is a triangular matrix, then the determinant of $A$ is the product of the elements on the leading diagonal

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