1. Introduction
Matrices are rectangular arrangements of numbers organized in rows and columns. They are powerful tools used to represent and manipulate data in science, economics, engineering, computer graphics, and real life decision-making. Operations on Matrices includes addition, subtraction, multiplication and division, just like arithmetic operations on numbers.
Example of a Matrix
A = | 2 4 |
| 1 3 |
- This is a 2 × 2 matrix
- 2 rows, 2 columns
2. Types of Matrices (Quick Review)
| Type | Description |
|---|---|
| Row Matrix | Only one row |
| Column Matrix | Only one column |
| Square Matrix | Rows = Columns |
| Zero Matrix | All elements are zero |
| Identity Matrix | 1s on the main diagonal |
Identity Matrix (2 × 2)
I = | 1 0 |
| 0 1 |
Operations on Matrices
A. Addition of Matrices
Two matrices can be added only if they have the same order.
Rule:
Add corresponding elements.
A = | 2 3 | B = | 1 4 |
| 5 6 | | 7 2 |
A + B = | 2+1 3+4 |
| 5+7 6+2 |
= | 3 7 |
|12 8 |
📌 Real-life idea: Adding daily sales from two shops.
B. Subtraction of Matrices
Same rule as addition, but subtract corresponding elements.
A − B = | 2−1 3−4 |
| 5−7 6−2 |
= | 1 −1 |
|−2 4 |
📌 Real-life idea: Comparing profits between two months.
C. Scalar Multiplication
Multiply each element by a number (scalar).
Let A = | 2 1 |
| 3 4 |
3A = | 6 3 |
| 9 12 |
📌 Real-life idea: Increasing production costs by a fixed rate.
D. Multiplication of Matrices
Matrix multiplication is not commutative:
AB ≠ BA (in most cases)
For matrix multiplication, the following conditions need to be met.
- Number of columns in first matrix = number of rows in second matrix.
A = | 1 2 | B = | 3 |
| 4 5 | | 6 |
AB = | (1×3 + 2×6) |
| (4×3 + 5×6) |
= | 15 |
| 42 |
📌 Real-life idea: Calculating total cost using quantity × price.
4. Real-Life Applications of Matrices
1. Economics & Business
Matrices are used to calculate profit, loss, production costs, and wages.
Cost Matrix × Quantity Matrix = Total Cost
2. Computer Graphics
Matrices are used to rotate, enlarge, and shrink images in video games and animation.
[Image Points] × [Transformation Matrix]
3. Transportation
Used to organize routes, distances, and schedules between cities.
4. Science & Engineering
Matrices model electrical circuits, population growth, and chemical reactions.
5. Worked Examples
Simple Example
Add:
A = | 2 1 |
| 3 4 |
B = | 5 6 |
| 1 2 |
Solution:
A + B = | 7 7 |
| 4 6 |
Complex Example
Multiply:
A = | 1 2 3 |
| 4 5 6 |
B = | 1 2 |
| 3 4 |
| 5 6 |
Solution:
AB = | (1×1+2×3+3×5) (1×2+2×4+3×6) |
| (4×1+5×3+6×5) (4×2+5×4+6×6) |
= | 22 28 |
| 49 64 |
6. Revision Exercise
(Arranged from simple to complex)
A. Basic Level
- Add:
| 3 2 | + | 1 4 |
| 5 1 | | 2 3 |
- Subtract:
| 7 5 | − | 4 2 |
| 6 3 | | 1 1 |
B. Intermediate Level
- If
A = | 2 3 |
| 1 4 |
Find 3A.
- Determine whether the following multiplication is possible:
(2 × 3) × (2 × 2)
C. Advanced Level
- Multiply:
| 1 0 2 | × | 3 |
| 2 1 1 | | 1 |
| 2 |
- If
A = | 1 2 |
| 3 4 |
Find A².
7. Application Questions (Real Life)
- A shop sells two products. The cost matrix is
| 50 30 |
and the quantity matrix is
| 10 |
| 20 |
Find the total cost.
- A school records boys and girls in two classes using matrices. How can matrix addition help find the total number of students?
- A graphics designer enlarges an image using a transformation matrix. Explain how scalar multiplication is applied.
- A transport company uses a matrix to represent distances between cities. How can matrix multiplication help calculate total travel distances?