Matrix Algorithms & Linear Algebra Basics

Linear algebra is about vectors and matrices. It helps us solve systems of equations. It’s useful in many areas of computer science.

1. What Is a Matrix?

A matrix is a table of numbers. It has rows and columns.

Example of a 2x2 matrix:

1	2
3	4

Rows go across. Columns go down.

2. Real-Life Example

Imagine you have 2 stores and you sell 2 products.

A matrix can show how many items each store sold:

	Product A	Product B
Store 1	5	8
Store 2	7	6

This is a 2x2 matrix. It helps us see and work with data easily.

3. Matrix Addition

You can add two matrices of the same size.

Example:

1	2
3	4

5	6
7	8

6	8
10	12

4. Matrix Multiplication

This is not like normal multiplication.

To multiply, rows of the first matrix are multiplied with columns of the second.

Example:

Matrix A (2x3):

1	2	3
4	5	6

Matrix B (3x2):

7	8
9	10
11	12

Result (2x2):

58	64
139	154

We multiplied row × column and added up the results.

5. Identity Matrix

This is like the number 1 for matrices.

1	0
0	1

Any matrix × identity = the same matrix.

6. Transpose

Flip rows and columns.

Example:

1	2
3	4

Transpose:

1	3
2	4

7. Determinant (2x2)

A number that tells if a matrix can be inverted.

For a 2x2 matrix:

a	b
c	d

Determinant = ad − bc

If it’s zero, we can’t find the inverse.

8. Inverse Matrix

Like dividing in matrix world.

If A × B = Identity, then B is the inverse of A.

Not all matrices have an inverse.

9. Solving Linear Equations

Example system:

2x + 3y = 8
x + y = 3

In matrix form:

A =

2	3
1	1

X =

B =

AX = B → we can solve it using inverse of A.

10. Applications in Real Life

Graphics: Used to rotate, scale, or move images.
AI/ML: Data is stored and processed as matrices.
Games: 3D models use linear algebra.
Maps: GPS systems use matrix math to locate you.

Summary

Matrices are tables of numbers.
You can add, multiply, and invert them.
They help solve systems of equations.
Used in many real-world tools: games, graphics, maps, AI.

Fun Fact

Pixar uses linear algebra to animate characters. Every movement you see involves matrix math!