How do you find the adjoint of a matrix?

Find every signed minor to form the cofactor matrix, then transpose that cofactor matrix.

Does a singular matrix have an adjoint?

Yes. Every square matrix has a classical adjoint, even when its determinant is zero and it has no inverse.

Adjoint of a Matrix Calculator With Steps

Q: Is adjoint the same as inverse?

No. If the determinant is nonzero, the inverse equals the adjoint divided by the determinant.

Calculate the classical adjoint (adjugate) of a 2 × 2 or 3 × 3 matrix. The result includes cofactors, the required transpose, the determinant, and the identity A·adj(A) = det(A)I.

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Adjoint formula

adj(A) = Cᵀ

C is the cofactor matrix. If det(A) ≠ 0, then A⁻¹ = adj(A) / det(A).

Adjoint Matrix Calculator

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What is the adjoint of a matrix?

The classical adjoint of a square matrix is the transpose of its cofactor matrix. It is also called the adjugate. This page uses “adjoint” in that classical elementary-matrix sense, not the conjugate transpose used for complex inner-product spaces.

To form a cofactor Cᵢⱼ, delete row i and column j, calculate the determinant of the remaining minor, and apply the sign (−1)ⁱ⁺ʲ. After calculating every cofactor, transpose the entire cofactor matrix.

Adjoint of a 2 × 2 matrix

For A = [[a, b], [c, d]], swap the diagonal entries and negate the off-diagonal entries:

adj(A) = [[d, −b], [−c, a]]

Example: if A = [[3, 2], [5, 4]], then adj(A) = [[4, −2], [−5, 3]]. The determinant is 3·4 − 2·5 = 2, so the inverse is one-half of that adjoint.

How to find the adjoint of a 3 × 3 matrix

Calculate the nine 2 × 2 minors.
Apply the checkerboard signs + − + / − + − / + − + to form the cofactor matrix.
Transpose: rows become columns.

The transpose step is essential. A cofactor matrix by itself is not generally the adjoint. The calculator shows both matrices side by side so this step remains visible.

How the calculator verifies adj(A)

Every square matrix satisfies A·adj(A) = adj(A)·A = det(A)I. The calculator multiplies A by its result and compares the product with det(A) times the identity matrix. This check works even for a singular matrix.

If det(A) is non-zero, the identity also proves the inverse formula A⁻¹ = adj(A)/det(A). If det(A) = 0, an adjoint still exists but division by the determinant is impossible, so no inverse exists.

Common adjoint mistakes

Forgetting the transpose. Cofactors form C; the adjoint is Cᵀ.

Using unsigned minors. Cofactor signs alternate in a checkerboard pattern beginning with plus at the top left.

Assuming a zero determinant means no adjoint. It means no inverse. The adjoint itself remains defined.

Confusing transpose with adjoint. Aᵀ only swaps rows and columns. adj(A) transposes the cofactor matrix, not A itself.

Frequently asked questions

How do I calculate the adjoint of a matrix?

Calculate each signed minor to form the cofactor matrix, then transpose it. The calculator displays both steps.

Is adjoint the same as inverse?

No. When det(A) is non-zero, divide adj(A) by det(A) to obtain the inverse.

Can a non-square matrix have a classical adjoint?

No. Minors, cofactors, and the classical adjugate used here require a square matrix.

Does this calculator accept fractions?

It accepts integers and decimal values. Convert a fraction such as 3/4 to 0.75 before entering it.

Related tools and reference

For eigenvalues and A = PDP⁻¹, use the matrix diagonalization calculator. To solve A𝐱 = b with row steps, use the simultaneous equations solver. For terminology and identities, see the Adjugate reference at Wolfram MathWorld.