Use this tool to compute the adjugate matrix (often written as adj(A)) for a square matrix. Enter values as decimals, integers, or fractions like 3/4.
What is an adj matrix?
In linear algebra, “adj matrix” usually means the adjugate matrix of a square matrix A, written as adj(A). The adjugate is built from cofactors, then transposed. It is especially useful for finding the inverse of a matrix when the determinant is non-zero.
The key identity is:
A-1 = adj(A) / det(A), whenever det(A) ≠ 0.
How to use this adj matrix calculator
- Select a square size (from 1×1 to 5×5).
- Enter all matrix values. Empty cells are treated as 0.
- Click Calculate adj(A).
- Read the determinant, adjugate matrix, and inverse (when it exists).
How the calculation works
1) Build minors and cofactors
For each entry aij, remove row i and column j to form a minor matrix. Compute its determinant and apply the checkerboard sign (-1)i+j. That gives the cofactor Cij.
2) Form the cofactor matrix
All cofactors are placed into a matrix C of the same size as A.
3) Transpose to get adj(A)
The adjugate is the transpose of the cofactor matrix: adj(A) = CT.
Why adjugates are useful
- Matrix inversion: Useful in theory and by-hand calculations.
- Proofs and derivations: Common in algebra and advanced calculus texts.
- Control systems and applied math: Shows up in symbolic manipulations.
- Education: Great for understanding determinants deeply.
Common mistakes to avoid
- Mixing up adjugate matrix and adjacency matrix (graph theory).
- Forgetting the cofactor sign pattern:
+ - + / - + - / + - +for 3×3. - Using adj(A)/det(A) when det(A) = 0 (inverse does not exist).
- Transcription errors when copying matrix entries.
Quick note on terminology
If you were looking for an adjacency matrix calculator for graphs, that is a different tool. This page computes adjugate matrices in linear algebra.