Chapter 2: Matrices

2.1 Basic Concepts of Matrices

A matrix is a rectangular array of numbers arranged in rows and columns, enclosed in brackets $[ ]$ or $( )$.

Order of a Matrix

The order (or dimension) of a matrix with $m$ rows and $n$ columns is written as $m \times n$ (read as "$m$ by $n$").

• Row Matrix: A matrix with only 1 row (order $1 \times n$).
• Column Matrix: A matrix with only 1 column (order $m \times 1$).
• Square Matrix: A matrix where the number of rows equals the number of columns ($m = n$).

Equal Matrices

Matrix $A$ is equal to matrix $B$ ($A = B$) if and only if:

1. Both matrices have the same order.
2. Every corresponding element in both matrices is identical ($a_{ij} = b_{ij}$).

2.2 Basic Operations on Matrices

Addition and Subtraction of Matrices

Matrices can only be added or subtracted if they have the same order. The operation is performed on corresponding elements.

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \pm \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a \pm e & b \pm f \\ c \pm g & d \pm h \end{pmatrix}$$

Scalar Multiplication

Multiplying a matrix by a scalar constant $k$ multiplies every individual element in the matrix by $k$:

$$k \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ka & kb \\ kc & kd \end{pmatrix}$$

Matrix Multiplication

Two matrices $A$ (order $m \times n$) and $B$ (order $p \times q$) can be multiplied to form $AB$ if and only if $n = p$ (the number of columns in $A$ equals the number of rows in $B$). The resulting matrix will have the order $m \times q$.

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix}$$

2.3 Identity Matrix and Inverse Matrix

Identity Matrix ($I$)

A square matrix with elements $1$ along the main diagonal and $0$ elsewhere. For any square matrix $A$, $AI = IA = A$.

$$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Inverse Matrix ($A^{-1}$)

If $A$ is a square matrix and there exists a matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$, then $A^{-1}$ is the inverse matrix of $A$.

For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:

• Determinant: $\det(A) = |A| = ad - bc$
• Inverse Formula: $$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, \quad \text{where } ad - bc \neq 0$$
• Singular Matrix: If $\det(A) = ad - bc = 0$, matrix $A$ does not have an inverse.

2.4 Solving Simultaneous Linear Equations using Matrices

A system of simultaneous equations:

$$\begin{aligned} ax + by &= p \\ cx + dy &= q \end{aligned}$$

can be expressed in matrix equation form $AX = B$ as:

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix}$$

To solve for $X = \begin{pmatrix} x \\ y \end{pmatrix}$, multiply both sides by $A^{-1}$ from the left:

$$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix}$$