A matrix is a grid of numbers. A matrix with m rows and n columns is called an m×n matrix. Matrices are fundamental to linear algebra, machine learning, and 3D graphics.
Key Operations
- Addition: Add corresponding elements (matrices must be same size)
- Multiplication: Row×column dot products (columns of A must equal rows of B)
- Determinant: Scalar value that indicates invertibility (det ≠ 0 means invertible)
- Transpose: Swap rows and columns (A^T)
2×2 Determinant
For matrix [[a,b],[c,d]]: det = ad − bc
Why Matrices Matter in Practice
Every 3D video game, CGI film, and CAD program uses matrices to rotate, scale, and move objects in space — a single matrix multiplication can transform thousands of points simultaneously. Search engines, recommendation systems, and neural networks also represent their core computations as large matrix operations, which is why GPUs (built for fast matrix math) power modern AI.
Identity and Inverse Matrices
The identity matrix (1s on the diagonal, 0s elsewhere) acts like the number 1 in matrix multiplication — multiplying any matrix by it leaves the original unchanged. A matrix's inverse (A⁻¹) is the matrix that, multiplied by the original, produces the identity matrix — used to solve systems of linear equations, but only exists when the determinant is non-zero.