Matrices

 

Matrices                           practice sums 👈

A matrix is a rectangular array of numbers arranged in rows and columns.

Example:

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

• It has 2 rows and 2 columns → a 2×2 matrix

---

2. Order of a Matrix

The order of a matrix is written as m × n, where:

• m = number of rows

• n = number of columns

Example:

$$\begin{bmatrix} 5 & 6 & 7 \end{bmatrix}$$

is a 1×3 matrix

---

3. Types of Matrices

Type	Example	Description
Row Matrix	$[1 \quad 2 \quad 3]$	Only 1 row
Column Matrix	$\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$	Only 1 column
Square Matrix	$\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$	Rows = Columns
Zero Matrix	$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$	All elements are 0
Diagonal Matrix	$\begin{bmatrix} 5 & 0 \\ 0 & 9 \end{bmatrix}$	Non-zero only on the diagonal
Identity Matrix	$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$	Diagonal = 1, others = 0

---

4. Matrix Notation

• Matrix is denoted by capital letters: A, B, C...

• Elements by $a_{i\mathrm{j\; where\; <em\; data-end="1751"\; data-start="1748">i \; \; is\; row\; number,</em><em\; data-end="1770"\; data-start="1767">j \; \; is\; column\; number</em>}}$

---

5. Matrix Operations

✅ Addition

• Add corresponding elements

$$A + B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$$

(Only possible if order is same)

✅ Subtraction

Same as addition, just subtract elements.

✅ Scalar Multiplication

Multiply each element by the number (scalar).

$$3 \times \begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 0 & 12 \end{bmatrix}$$

✅ Matrix Multiplication

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

(Only possible if: number of columns in A = number of rows in B)