About

The adjoint (or adjugate) of a square matrix A is the transpose of its cofactor matrix. It appears directly in the formula for the matrix inverse: A⁻¹ = 1det(A) ⋅ adj(A). An error in computing any single cofactor propagates through the entire adjugate and corrupts the inverse. This tool computes the minor for every element, applies the sign factor (−1)^i+j, transposes to produce adj(A), and returns the determinant det(A) as a side result. It handles matrices from 2×2 up to 6×6.

The computation uses Laplace cofactor expansion recursively. Floating-point rounding is applied at display time to 6 significant digits. Note: for singular matrices (det(A) = 0), the adjoint still exists but the inverse does not. This tool approximates real-valued entries; it does not handle symbolic or complex inputs.

Formulas

The adjoint of an n × n matrix A is computed in three steps.

Step 1. For each element a_ij, compute the minor M_ij as the determinant of the submatrix formed by deleting row i and column j.

Step 2. Compute the cofactor:

C_ij = (−1)^i+j ⋅ M_ij

Step 3. The adjoint is the transpose of the cofactor matrix:

adj(A) = C^T

That is, the element at row i, column j of the adjoint equals C_ji.

The determinant is computed via Laplace expansion along the first row:

det(A) = n∑j=1 a_1j ⋅ C_1j

For a 2×2 matrix, the adjoint has a closed form:

adj abcd = d−b−ca

Where a, b, c, d are the elements of the original 2×2 matrix. M_ij = minor of element at row i, column j. C_ij = cofactor of element at row i, column j. n = dimension of the square matrix.

Reference Data

Matrix Size	Number of Minors	Minor Size	Cofactor Sign Pattern (Row 1)	Max Recursive Depth
2×2	4	1×1	+ −	1
3×3	9	2×2	+ − +	2
4×4	16	3×3	+ − + −	3
5×5	25	4×4	+ − + − +	4
6×6	36	5×5	+ − + − + −	5
Key Properties of the Adjoint Matrix
Identity relation		A ⋅ adj(A) = det(A) ⋅ I
Adjoint of identity		adj(I) = I
Adjoint of transpose		adj(A^T) = (adj(A))^T
Adjoint of product		adj(AB) = adj(B) ⋅ adj(A)
Determinant of adjoint		det(adj(A)) = (det(A))ⁿ⁻¹
Scalar multiple		adj(kA) = kⁿ⁻¹ adj(A)
Adjoint of adjoint		adj(adj(A)) = (det(A))ⁿ⁻² ⋅ A (for n ≥ 2)
Rank when det = 0		rank(adj(A)) ≤ 1
Inverse formula		A⁻¹ = 1det(A) adj(A), det(A) ≠ 0

Frequently Asked Questions

The adjoint (adjugate) is the transpose of the cofactor matrix and always exists for any square matrix. The inverse exists only when the determinant is non-zero. The inverse is computed as A⁻¹ = (1 / det(A)) ⋅ adj(A). For singular matrices (det(A) = 0), the adjoint still exists but the inverse does not.

For matrices up to 6×6, IEEE 754 double-precision provides approximately 15-16 significant digits, which is more than sufficient. Errors compound with each recursive determinant call, but at n ≤ 6 the maximum recursion depth is 5, keeping accumulated error well below 10⁻¹⁰ for typical inputs. Results are displayed rounded to 6 significant figures.

Yes. If the rank of A is less than n − 1 (where n is the matrix dimension), then every (n − 1) × (n − 1) minor is zero. This means the entire cofactor matrix is zero, and therefore adj(A) is the zero matrix. For example, any 3×3 matrix of rank 1 or 0 has a zero adjoint.

Laplace cofactor expansion is the natural algorithm when you need every individual cofactor (not just the determinant). LU decomposition computes the determinant in O(n³) but does not directly yield the cofactor matrix. Since this tool must output all n² cofactors, and n ≤ 6, the factorial complexity of Laplace expansion is acceptable (at most 720 operations per minor).

Yes. For any square matrix A, both A ⋅ adj(A) and adj(A) ⋅ A equal det(A) ⋅ I, where I is the identity matrix. This holds regardless of whether A is invertible.

The calculator validates every cell before computing. Non-numeric entries, empty cells, or special values like Infinity and NaN are rejected with a specific error toast indicating which cell is invalid. The calculation will not proceed until all entries are valid real numbers.