Adjoint Matrix Calculator — Adjugate of 2×2 and 3×3 Matrices
Enter a 2×2 or 3×3 matrix to get its adjoint (adjugate) and determinant instantly, with the cofactor steps shown so you can follow the working.
Matrix size
det(A) — non-zero means the matrix is invertible and the adjoint is valid
Compute determinant: det(A) = ad − bc
Swap main-diagonal entries (a↔d), negate off-diagonal (b→−b, c→−c)
- 1
Product of main diagonal: a × d
1 × 1 = 1 - 2
Product of anti-diagonal: b × c
2 × 0 = 0 - 3
Determinant: ad − bc
1 − 0 = 1
How does this calculator work?
The adjoint (adjugate) of A is the transpose of its cofactor matrix. For 2×2 it is simply [[d,−b],[−c,a]]; for 3×3 each cofactor requires a 2×2 minor determinant with alternating signs. Dividing the adjoint by det(A) yields the matrix inverse whenever det(A) ≠ 0.
Formula
How this is calculated
The adjoint (or adjugate) of a square matrix A is the transpose of its cofactor matrix. Each cofactor C(i,j) is computed by removing row i and column j from A, taking the determinant of the remaining sub-matrix (the "minor"), and multiplying by (−1)^(i+j) to apply the checkerboard sign pattern. The adjoint is formed by placing C(j,i) in position (i,j) — that is, transposing the cofactor matrix.
For a 2×2 matrix [[a,b],[c,d]], the formula simplifies to a direct swap: adj = [[d,−b],[−c,a]] and det = ad − bc. For 3×3 matrices, all nine cofactors must be computed and the determinant is found by expanding along the first row.
The adjoint is most important because it gives the inverse: A⁻¹ = adj(A) / det(A). If the determinant is zero the matrix is singular and has no inverse, but the adjoint itself is still defined. In numerical applications the adjoint is used in Cramer's rule and to derive analytical inverse formulas without numerical division.
Frequently asked questions
The cofactor matrix C has C(i,j) at position (i,j). The adjoint (adjugate) is its transpose: adj(A)[i][j] = C(j,i). Many texts use the terms interchangeably but the transpose step is required to get the adjoint.
Divide every entry of adj(A) by the determinant: A⁻¹ = adj(A) / det(A). This only works when det(A) ≠ 0 (the matrix is invertible). If det(A) = 0, the matrix is singular and has no inverse.
This calculator handles 2×2 and 3×3 matrices. For larger matrices the same cofactor-transpose rule applies but the number of minors grows as n², making it computationally expensive — numerical methods such as LU decomposition are preferred for sizes 4×4 and above.
Also known as
TG we-Calculate Editorial Team. (2026). Adjoint Matrix Calculator — Adjugate of 2×2 and 3×3 Matrices [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/adjoint-matrix-calculator
TG we-Calculate Editorial Team. "Adjoint Matrix Calculator — Adjugate of 2×2 and 3×3 Matrices." TG we-Calculate. 2026. https://we-calculate.com/calculator/adjoint-matrix-calculator.
TG we-Calculate Editorial Team, "Adjoint Matrix Calculator — Adjugate of 2×2 and 3×3 Matrices," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/adjoint-matrix-calculator
@misc{wecalculate_adjoint_matrix_calculator, title = {Adjoint Matrix Calculator — Adjugate of 2×2 and 3×3 Matrices}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/adjoint-matrix-calculator}}, year = {2026}, note = {TG we-Calculate} }
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