Matrix Operations
Reference for matrix addition, subtraction, multiplication, determinant, inverse, and transpose.
Includes 2x2 and 3x3 worked examples for linear algebra.
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
Common Mistakes
- Matrix multiplication is not commutative: AB ≠ BA in general — reversing the order usually produces an entirely different matrix, not just a different arrangement
- For A × B to be defined, columns of A must equal rows of B; the result has dimensions (rows of A) × (columns of B) — mismatched dimensions is the most common beginner error
- A matrix inverse exists only when det(A) ≠ 0; a singular matrix (det = 0) has no inverse and represents a system with no unique solution or infinitely many solutions
- Transpose of a product reverses order: (AB)ᵀ = BᵀAᵀ — forgetting this reversal when transposing chained multiplications is a frequent mistake in proofs and derivations