Adjugate Matrix Calculator
Compute the adjugate (classical adjoint) of a square matrix. The adjugate is the transpose of the cofactor matrix and is used to find the matrix inverse.
Notes
What is the Adjugate?
The adjugate (classical adjoint) of A, written adj(A), is the transpose of the cofactor matrix C. It satisfies the identity A · adj(A) = det(A) · I.
2×2 Adjugate
For A = [[a,b],[c,d]], the cofactor matrix is [[d,−c],[−b,a]] and the adjugate (its transpose) is [[d,−b],[−c,a]]:
Example: A = [[3,1],[2,4]]. Cofactors: C₁₁=4, C₁₂=−2, C₂₁=−1, C₂₂=3. adj(A):
| 4 | -1 |
| -2 | 3 |
det(A) = 12−2 = 10. So A⁻¹ = (1/10) · [[4,−1],[−2,3]].
- Matrix Calculations Guide — Cofactor Matrix & Adjugate — In-depth notes on Cofactor Matrix & Adjugate with worked examples
Frequently Asked Questions
What is the difference between adjugate and adjoint?
In linear algebra, 'adjugate' refers to the transpose of the cofactor matrix. In functional analysis, 'adjoint' means the conjugate transpose. They are different concepts.
Can I use the adjugate to find the inverse?
Yes: A⁻¹ = adj(A) / det(A), provided det(A) ≠ 0.
What is adj(adj(A))?
For an n×n matrix: adj(adj(A)) = det(A)^(n-2) · A. For 2×2 matrices this means adj(adj(A)) = A.