Adjugate Matrix Calculator
Build the cofactor matrix from minors, transpose it into the adjugate, and see both A·adj(A) = det(A)·I and A⁻¹ = adj(A)/det(A) verified entry by entry.
Cofactor matrix C
| 24 | 5 | -4 |
| -12 | 3 | 2 |
| -2 | -5 | 4 |
Cᵢⱼ = (−1)i+j·Mᵢⱼ, where Mᵢⱼ is the determinant of A with row i and column j deleted.
Adjugate adj(A) = Cᵀ
| 24 | -12 | -2 |
| 5 | 3 | -5 |
| -4 | 2 | 4 |
The adjugate (classical adjoint) is the transpose of the cofactor matrix — swap rows and columns.
Identity 1 — A · adj(A) = det(A)·I
| 22 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
Every diagonal entry equals det(A) = 22 and every off-diagonal entry is 0, exactly as the identity predicts. This holds for singular matrices too — then both sides are the zero matrix.
Identity 2 — A⁻¹ = adj(A) / det(A)
| 12/11 | -6/11 | -1/11 |
| 5/22 | 3/22 | -5/22 |
| -2/11 | 1/11 | 2/11 |
Every entry of the adjugate divided by det(A) = 22. This is the closed-form inverse taught alongside Gauss-Jordan; it is exact but grows expensive fast, because an n×n adjugate needs n² determinants of size (n−1).
Entry-by-entry working
| Position | Minor Mᵢⱼ | Sign | Cofactor Cᵢⱼ | Lands at |
|---|---|---|---|---|
| (1, 1) | 24 | + | 24 | adj(A)[1, 1] |
| (1, 2) | -5 | − | 5 | adj(A)[2, 1] |
| (1, 3) | -4 | + | -4 | adj(A)[3, 1] |
| (2, 1) | 12 | − | -12 | adj(A)[1, 2] |
| (2, 2) | 3 | + | 3 | adj(A)[2, 2] |
| (2, 3) | -2 | − | 2 | adj(A)[3, 2] |
| (3, 1) | -2 | + | -2 | adj(A)[1, 3] |
| (3, 2) | 5 | − | -5 | adj(A)[2, 3] |
| (3, 3) | 4 | + | 4 | adj(A)[3, 3] |
What is the Adjugate Matrix Calculator?
The ByteTools Adjugate Matrix Calculator computes the classical adjoint of a square matrix from 2×2 up to 5×5.
- Cofactor matrix and adjugate for 2×2 through 5×5 matrices
- A·adj(A) = det(A)·I multiplied out and verified on screen
- A⁻¹ = adj(A)/det(A) computed when the determinant is non-zero
- Entry-by-entry table of every minor, sign, cofactor and its destination
- Exact fraction arithmetic throughout
- Fully client-side — nothing uploaded, works offline
How to use the Adjugate Matrix Calculator
- 1
Enter a square matrix, one row per line, values separated by spaces or commas.
- 2
Read the cofactor matrix C and the adjugate adj(A) = Cᵀ side by side.
- 3
Check the A·adj(A) panel, where every diagonal entry should equal det(A).
- 4
Read the inverse panel, where each adjugate entry is divided by det(A).
- 5
Scroll to the working table to see each minor, sign and cofactor individually.
About the Adjugate Matrix Calculator
The ByteTools Adjugate Matrix Calculator computes the classical adjoint of a square matrix from 2×2 up to 5×5. It builds each cofactor as Cᵢⱼ = (−1)^(i+j)·Mᵢⱼ, where Mᵢⱼ is the determinant of the matrix with row i and column j deleted, then transposes the whole cofactor matrix to give adj(A).
Two identities are demonstrated with the actual numbers, not just stated. A·adj(A) is multiplied out and shown to be det(A) times the identity matrix — a result that holds even for singular matrices, where both sides come out as zero. And when det(A) is non-zero, dividing the adjugate by it gives the inverse, which is the closed-form alternative to Gauss-Jordan elimination.
A full working table lists every position with its minor, its sign, the resulting cofactor and where it lands after transposition, so you can check your own arithmetic line by line. Everything is computed in exact fractions by JavaScript in your browser, with no uploads and no stored data.
Frequently asked questions
What is the adjugate of a matrix?
The adjugate, also called the classical adjoint, is the transpose of the cofactor matrix. Its defining property is that A times its adjugate equals the determinant of A times the identity matrix, which is what makes it the route to a closed-form inverse.
What is the difference between adjugate and adjoint?
In older textbooks 'adjoint' means the adjugate described here. In modern usage, adjoint usually means the conjugate transpose from inner-product spaces, which is a different thing entirely. 'Adjugate' is the unambiguous term for the cofactor transpose.
How do you find the inverse using the adjugate?
Divide every entry of the adjugate by the determinant: A⁻¹ = adj(A)/det(A). It is exact and closed-form, but expensive — an n×n adjugate needs n² determinants of size n−1, so Gauss-Jordan elimination is far faster for large matrices.
What is the adjugate of a singular matrix?
It still exists and is still well defined; you just cannot divide by the determinant to get an inverse. The identity A·adj(A) = det(A)·I still holds, with both sides equal to the zero matrix.
How do you find the cofactor of a matrix element?
Delete the element's row and column, take the determinant of what is left — that is the minor — and multiply by (−1) raised to the sum of the row and column indices. The signs alternate in a checkerboard pattern starting with + in the top-left corner.
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