The adjugate packages all the cofactors of a matrix. Its key property, A · adj(A) = det(A) · I, gives a formula for the inverse and for Cramer’s rule. For a 2 × 2 matrix [[a, b], [c, d]], adj(A) = [[d, −b], [−c, a]].
How to use the Adjoint Matrix Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
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How it works
Each cofactor is a signed minor – the determinant of the matrix with one row and one column removed. The cofactor matrix is transposed to give the adjugate; dividing by the determinant gives the inverse.
Formula
adj(A) = Cᵀ, where Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ
A⁻¹ = adj(A) / det(A)
Worked example
Inputs — Matrix size: 3 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; a₃: 2; b₃: 0; c₃: 5.
Results — Adjoint (adjugate) matrix = ⎡ 15 −5 −6 ⎤
⎢ −5 16 2 ⎥
⎣ −6 2 11 ⎦; Determinant = 43; Inverse A⁻¹ = adj(A)/det(A) = ⎡ 15/43 −5/43 −6/43 ⎤
⎢ −5/43 16/43 2/43 ⎥
⎣ −6/43 2/43 11/43 ⎦.
Key terms
- Cofactor
- A signed minor (−1)^(i+j) Mᵢⱼ.
- Adjugate
- The transpose of the cofactor matrix.
Frequently asked questions
What is the adjoint of [[a, b], [c, d]]?
[[d, −b], [−c, a]].
Is the adjoint the same as the adjugate?
In this context yes; "adjoint" also means the conjugate transpose in other contexts.
What if det(A) = 0?
The adjugate still exists, but the matrix has no inverse.
Why transpose the cofactor matrix?
So that A · adj(A) = det(A) · I.