🔢 Adjoint Matrix Calculator
Calculate the adjoint (adjugate) matrix, cofactor matrix, and inverse
How to Use This Calculator
Select Matrix Size
Choose the size of your square matrix (2×2, 3×3, or 4×4).
Enter Matrix Elements
Input all elements of your square matrix. Enter numbers only (decimals are allowed).
Click Calculate
Press "Calculate Adjoint Matrix" to compute the adjoint, cofactor matrix, determinant, and inverse (if it exists).
View Results
Review the cofactor matrix, adjoint matrix, determinant, and inverse matrix (if determinant ≠ 0).
Formula
adj(A) = (Cofactor Matrix)ᵀ
A⁻¹ = adj(A) / det(A) (if det(A) ≠ 0)
Steps to Calculate Adjoint:
- Calculate the cofactor matrix C, where Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ
- Mᵢⱼ is the minor (determinant of submatrix after removing row i and column j)
- Transpose the cofactor matrix to get adj(A)
Example: 2×2 Matrix
For matrix A = [a b; c d]
Cofactor matrix = [d -c; -b a]
Adjoint = [d -b; -c a] (transpose)
det(A) = ad - bc
A⁻¹ = [d -b; -c a] / (ad - bc)
About Adjoint Matrix Calculator
The Adjoint Matrix Calculator computes the adjoint (also called adjugate) of a square matrix. The adjoint is the transpose of the cofactor matrix and is essential for finding the inverse of a matrix: A⁻¹ = adj(A) / det(A).
When to Use This Calculator
- Linear Algebra: Find matrix inverses and solve systems of equations
- Engineering: Solve linear systems in circuit analysis, control theory
- Computer Graphics: Transform matrices and rotations
- Cryptography: Matrix operations in encryption algorithms
- Economics: Input-output models and matrix operations
Why Use Our Calculator?
- ✅ Complete Solution: Shows cofactor matrix, adjoint, determinant, and inverse
- ✅ Multiple Sizes: Supports 2×2, 3×3, and 4×4 matrices
- ✅ Step-by-Step: Clear display of all intermediate results
- ✅ Accurate: Precise calculations with decimal support
- ✅ Educational: Helps understand matrix theory
- ✅ Free: No registration required
Key Concepts
- Cofactor: Cᵢⱼ = (-1)ⁱ⁺ʲ × det(Mᵢⱼ), where Mᵢⱼ is the minor matrix
- Minor: Determinant of the submatrix obtained by removing row i and column j
- Adjoint: Transpose of the cofactor matrix
- Inverse: Only exists if determinant ≠ 0 (non-singular matrix)
- Property: A × adj(A) = adj(A) × A = det(A) × I
Applications
Solving Linear Systems: For Ax = b, if A is invertible, then x = A⁻¹b = (adj(A) / det(A)) × b.
Matrix Decomposition: Adjoint is used in various matrix factorization methods.
Frequently Asked Questions
What is an adjoint matrix?
The adjoint (adjugate) of a matrix A is the transpose of its cofactor matrix. It's used to compute the inverse: A⁻¹ = adj(A) / det(A).
What's the difference between adjoint and cofactor matrix?
The cofactor matrix contains cofactors Cᵢⱼ. The adjoint is simply the transpose of the cofactor matrix: adj(A) = (Cofactor Matrix)ᵀ.
Can I use adjoint to find the inverse?
Yes! If det(A) ≠ 0, then A⁻¹ = adj(A) / det(A). If det(A) = 0, the matrix is singular and has no inverse.
What is a cofactor?
A cofactor Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ, where Mᵢⱼ is the minor (determinant of the submatrix after removing row i and column j).
Why does my inverse not exist?
If the determinant is zero, the matrix is singular and has no inverse. Only non-singular (invertible) matrices have inverses.
Does adj(A) equal adj(Aᵀ)?
No, but adj(Aᵀ) = (adj(A))ᵀ. The adjoint of a transpose is the transpose of the adjoint.
What's the relationship between adjoint and determinant?
A × adj(A) = adj(A) × A = det(A) × I, where I is the identity matrix. This is the fundamental property linking adjoint and determinant.