MATRICES

Rectangular arrays of numbers. The fundamental data structure for storing linear equations, pixels, and complex transformations.

01 // Structure

A Matrix is defined by its dimensions: Rows (m) × Columns (n). Each number inside is an element with a specific coordinate address $(i, j)$ , allowing computers to instantly access massive grids of data.

m Rows

n Columns

A = \begin{bmatrix} 1 & 0 & 4 \\ 2 & 5 & 9 \end{bmatrix}

Dimensions:

2 \times 3

02 // The Crash (Multiplication)

Matrix multiplication isn't just scaling numbers—it is applying a set of spatial transformations. To multiply matrices, you crash the Row of the first matrix into the Column of the second, calculating the dot product.

The Multiplication Engine

Matrix A (2×2)

Vector B (2×1)

Result C (2×1)

Dot Product Breakdown

Row 1 • Col 1 = C₁

(1 × 5) + (2 × 6)

5 + 12 = 17

Row 2 • Col 1 = C₂

(3 × 5) + (4 × 6)

15 + 24 = 39

03 // Special Matrices

The One

Identity (I)

A square matrix with 1s on the diagonal and 0s elsewhere. Multiplying by $I$ does absolutely nothing to the original matrix.

A \cdot I = A

The Flip

Transpose (

A^T

)

Swap the rows and columns. Row 1 becomes Column 1. This reflects the entire matrix perfectly over its main diagonal.

\begin{bmatrix} 1 & 2 \end{bmatrix}^T = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

The Undo

Inverse (

A^{-1}

)

The matrix that "undoes" A. Only square matrices with non-zero determinants possess one.

A \cdot A^{-1} = I

Grid Logic Initialized

You are ready to command computational systems.

Next: Solvers