What is the adjugate of a matrix?

The adjugate (classical adjoint) of a square matrix A is the transpose of the cofactor matrix of A. Each entry adj(A)(i,j) equals the cofactor C(j,i) of A.

How is the adjugate used to find the inverse?

If det(A) is non-zero, then A⁻¹ = adj(A) / det(A). This tool computes the adjugate; divide each entry by the determinant to get the inverse.

The cofactor C(i,j) of entry a(i,j) is (-1)^(i+j) multiplied by the determinant of the submatrix formed by deleting row i and column j.

Does the tool work for singular (non-invertible) matrices?

Yes. The adjugate is defined for any square matrix regardless of whether it is invertible. The tool will compute it even when det(A) = 0.

Can I use decimal or negative values?

Yes. Any real number (including decimals and negatives) is accepted in each cell.

What is the adjugate of a 1x1 matrix?

For a 1x1 matrix [a], the adjugate is [1] by convention. This tool covers 2x2 and 3x3 cases.

Matrix Adjugate Calculator

Enter a 2x2 or 3x3 matrix and instantly get its adjugate (classical adjoint), cofactor matrix, and step-by-step working.

Matrix Input

Matrix Size

Matrix A

Enter a matrix and click Compute Adjugate.

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Summary

The adjugate (or classical adjoint) of a square matrix is the transpose of its cofactor matrix. It is used to compute the matrix inverse via the formula A⁻¹ = adj(A) / det(A), and appears throughout linear algebra and engineering mathematics. This tool accepts a 2x2 or 3x3 matrix, computes each cofactor, assembles the cofactor matrix, transposes it to get the adjugate, and shows every intermediate step.

How it works

Select the matrix size — 2x2 or 3x3.
Enter the matrix entries in the grid.
Click Compute to calculate the adjugate.
The tool displays each cofactor C(i,j) with its minor determinant.
The cofactor matrix is transposed to produce the adjugate.
Use Reset to clear all entries and start again.

Use cases

Computing matrix inverses without a computer algebra system.
Verifying hand-worked cofactor and adjugate calculations.
Studying linear algebra — seeing each cofactor step-by-step.
Solving systems of linear equations via Cramer's rule.
Engineering and physics calculations involving matrix inverses.
Checking textbook adjugate examples instantly.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu