Adjoint (Adjugate) Matrix Calculator (adj A from cofactors, with A⁻¹)

Understanding the adjoint (adjugate) matrix calculator — concept and principle

This calculator finds the adjugate, also called the classical adjoint, of a square matrix from its cofactors. It also shows the determinant, the inverse when it exists, and the check A·adj(A) = det(A)·I. Use it for linear algebra homework, Cramer's rule or inverting small matrices by hand.

Each entry is a signed 2 × 2 minor; dividing adj(A) by det A gives the inverse.

How it is calculated

Minor M_ij = A with row i and column j removed

Cofactor C_ij = (−1)^(i+j) × det(M_ij)

adj(A) = transpose of the cofactor matrix: adj(A)_ij = C_ji

A·adj(A) = adj(A)·A = det(A)·I

If det(A) ≠ 0: A⁻¹ = adj(A) ÷ det(A)

For a 2 × 2 matrix [a b; c d], adj = [d −b; −c a].
Entries can be integers, decimals or fractions; whole-number matrices give exact results.

The principle

Expanding a determinant along row i gives det(A) = Σ a_ik C_ik. If you instead pair row i with the cofactors of a different row, you get the determinant of a matrix with two equal rows, which is 0. Putting the cofactors in transposed positions arranges exactly these products, so A·adj(A) has det(A) on the diagonal and zeros elsewhere. That is why dividing the adjugate by the determinant gives the inverse. Unlike the inverse, the adjugate exists for every square matrix, even a singular one.

Worked example

Default input: A = [1 −1 2; 1 1 1; 1 3 −1].

C₁₁ = +det[1 1; 3 −1] = −1 − 3 = −4; C₁₂ = −det[1 1; 1 −1] = −(−2) = 2; C₁₃ = +det[1 1; 1 3] = 2.
Row 2: C₂₁ = −det[−1 2; 3 −1] = 5; C₂₂ = +det[1 2; 1 −1] = −3; C₂₃ = −det[1 −1; 1 3] = −4.
Row 3: C₃₁ = +det[−1 2; 1 1] = −3; C₃₂ = −det[1 2; 1 1] = 1; C₃₃ = +det[1 −1; 1 1] = 2.
Transposing: adj(A) = [−4 5 −3; 2 −3 1; 2 −4 2].
det(A) = 1·(−4) + (−1)·2 + 2·2 = −2, so A·adj(A) = [−2 0 0; 0 −2 0; 0 0 −2] and A⁻¹ = [2 −5/2 3/2; −1 3/2 −1/2; −1 2 −1]. Size 3 × 3.

How to read the result

If det(A) = 0 the matrix has no inverse, but the adjugate still exists. Then A·adj(A) is the zero matrix. The cofactor method is practical by hand up to 3 × 3; for larger matrices elimination is much faster, though the adjugate remains useful in proofs and in Cramer's rule.

Common mistakes

Forgetting to transpose the cofactor matrix.
Getting the checkerboard signs (−1)^(i+j) wrong.
Confusing the adjugate with the conjugate transpose, which is also sometimes called the adjoint.
Dividing by the determinant when it is 0.

Keep in mind: Large or nearly singular matrices can give large rounding errors in decimal form; prefer exact fractions where possible.

Frequently asked questions

What is the adjugate of a 2 × 2 matrix?

Swap the diagonal entries and change the sign of the off-diagonal ones: [d −b; −c a].

Is the adjoint the same as the inverse?

No. The inverse is the adjugate divided by the determinant, and only exists when the determinant is not 0.

Does a singular matrix have an adjugate?

Yes, every square matrix does.

Sources: A. Davis, P. Zachlin et al., Coordinated Linear Algebra (University of Nebraska–Lincoln), §5.3 · D. Margalit & J. Rabinoff, Interactive Linear Algebra, §4.2 Cofactor Expansions

Checked 9 Nov 2026. Coordinated Linear Algebra (A. Davis, P. Zachlin et al., University of Nebraska–Lincoln), §5.3 Extra Topic: Cramer’s Rule — adjugate as the matrix of cofactors transposed and Theorem 5.3.4 A⁻¹ = adj(A)/det A (Example 5.3.5: det = −2); D. Margalit & J. Rabinoff, Interactive Linear Algebra, §4.2 Cofactor Expansions, Theorem 4.2.2.

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