Matrix Operations
Reference for matrix addition, multiplication, determinant, and inverse formulas with worked examples.
The Formulas
Addition: [A + B]ᵢⱼ = aᵢⱼ + bᵢⱼ
Multiplication (2×2): AB where (AB)ᵢⱼ = Σ aᵢₖ × bₖⱼ
Determinant (2×2): det(A) = ad - bc for A = [[a,b],[c,d]]
Inverse (2×2): A⁻¹ = (1/det(A)) × [[d,-b],[-c,a]]
Multiplication (2×2): AB where (AB)ᵢⱼ = Σ aᵢₖ × bₖⱼ
Determinant (2×2): det(A) = ad - bc for A = [[a,b],[c,d]]
Inverse (2×2): A⁻¹ = (1/det(A)) × [[d,-b],[-c,a]]
Matrix operations are the building blocks of linear algebra. They are used in computer graphics, physics simulations, data science, and engineering.
Variables
| Symbol | Meaning |
|---|---|
| A, B | Matrices |
| aᵢⱼ | Element in row i, column j of matrix A |
| det(A) | Determinant of matrix A |
| A⁻¹ | Inverse of matrix A (only exists if det(A) ≠ 0) |
Example 1
Find the determinant of A = [[3, 7], [1, 5]]
det(A) = (3)(5) - (7)(1) = 15 - 7
= 8
Example 2
Find the inverse of A = [[4, 2], [1, 3]]
det(A) = (4)(3) - (2)(1) = 12 - 2 = 10
A⁻¹ = (1/10) × [[3, -2], [-1, 4]]
= [[0.3, -0.2], [-0.1, 0.4]]
When to Use Them
Use matrix operations when:
- Solving systems of linear equations
- Performing transformations in computer graphics (rotation, scaling)
- Working with data in machine learning and statistics
- Modeling networks, circuits, or any multi-variable system