Matrix Algebra: The Engine of Modern Computing
A matrix (plural: matrices) is a rectangular array of numbers arranged in rows and columns. While they might look like simple spreadsheet grids, matrices are the fundamental objects used to describe linear transformations in space. They are the reason you can rotate a 3D character in a video game or filter an image in Photoshop.
1. Basic Operations
Just like numbers, you can add, subtract, and multiply matrices—but with specific rules.
Addition & Subtraction
You can only add matrices if they have the exact same dimensions. You simply add the corresponding entries.
Multiplication (The Tricky Part)
Matrix multiplication is not element-by-element. It is a "Row by Column" dot product. To find the entry in the 1st row and 1st column of the answer, you multiply the entire 1st row of Matrix A by the entire 1st column of Matrix B and sum them up.
If A is 2x3, B must be 3xN.
Order matters! (Usually)
2. Determinants & Inverses
The Determinant is a single number that tells you if a matrix squishes space into a lower dimension. If Det = 0, the matrix has no inverse (it destroys information).
3. Real World Applications
- Computer Graphics: Every movement in a 3D game is a matrix calculation. A "Translation Matrix" moves you, a "Rotation Matrix" turns you, and a "Projection Matrix" flattens the 3D world onto your 2D screen.
- Cryptography: Matrices are used to scramble data (encryption). You need the "Inverse Matrix" (the key) to unscramble it.
- Economics: "Input-Output Models" use giant matrices to represent interactions between different industries in an economy.