Adjoint Matrix Calculator (Adjugate Matrix) — Find adj(A) of a 2×2, 3×3 or 4×4 Matrix

Adjoint Matrix Calculator (Adjugate Matrix)

Welcome to the adjoint matrix calculator! Here you can quickly and easily determine the adjoint (a.k.a. adjugate) of a square matrix. Not sure what the adjoint of a matrix is? Need a quick reminder of how to find the adjoint of a matrix? In the article below we teach you all you need to know about the adjoint/adjugate matrix! We give the adjugate matrix formula and explain how to find the adjugate matrix of size 2×2 via this formula.

This calculator handles any square matrix from 2×2 up to 4×4. It shows you the cofactor of every entry, the transpose step that turns the cofactor matrix into the adjugate, the determinant, the identity A·adj(A) = det(A)·I, and optionally the full inverse.

What is the adjoint of a matrix (adjugate matrix)?

First of all, be aware that what we call the adjoint matrix here is sometimes called the adjugate matrix. You may also encounter the term classical adjoint matrix. This confusion stems from the fact that, in some contexts, the term adjoint can mean the conjugate transpose of a matrix, which is something entirely different from what we consider here. We will freely mix the terms adjoint and adjugate so that you could quickly get used to both of them.

How to find the adjoint of a matrix?

Suppose A is an n × n matrix with real or complex entries. To find the adjugate of A, follow these steps:

1. Delete the i-th row and the j-th column of A. What you get is a (n − 1) × (n − 1) submatrix of A.
2. Compute the determinant of this submatrix. What you get is called the (i, j)-minor of A.
3. Multiply the (i, j)-minor of A by the sign factor (−1)^i+j. What you get is called the (i, j)-cofactor of A.
4. Repeat Steps 1, 2 and 3 for all i, j = 1, ..., n.
5. The adjoint of A is the n × n matrix whose (i, j) entry is the (j, i) cofactor of A. Note that the indices are flipped!

👉 The adjoint of matrix A is often denoted by adj(A).

🔎 If you are already familiar with the notion of cofactor matrix, then you may have realized that adj(A) is in fact the transpose of the cofactor matrix of A:

adj(A) = cof(A)T,   where   Cij = (−1)i+j · Mij

Adjugate of a 2×2 matrix

Let's see how the adjugate matrix formula explained above works in the simplest case. Namely, we will use it to find the adjugate of a 2×2 matrix. Consider the following matrix:

A = ⎡ a  b ⎤
    ⎣ c  d ⎦

There are four coefficients, so we repeat Steps 1, 2 and 3 four times:

• Deleting row 1 and column 1 leaves the single coefficient d. The sign factor is (−1)¹⁺¹ = +1, so the (1, 1)-cofactor is d.
• Deleting row 1 and column 2 leaves c. The sign factor is (−1)¹⁺² = −1, so the (1, 2)-cofactor is −c.
• Deleting row 2 and column 1 leaves b. The sign factor is (−1)²⁺¹ = −1, so the (2, 1)-cofactor is −b.
• Deleting row 2 and column 2 leaves a. The sign factor is (−1)²⁺² = +1, so the (2, 2)-cofactor is a.

Next, to find the adjugate we put the (i, j)-cofactor into the j-th row and i-th column (that's the index flip). This gives the formula worth memorizing:

adj(A) = ⎡  d  −b ⎤
         ⎣ −c   a ⎦

In words: swap the entries on the main diagonal (a and d), and flip the signs of the off-diagonal entries (b and c).

How to use this adjugate matrix calculator

1. Pick the matrix size — 2×2, 3×3, or 4×4. Only square matrices have an adjugate.
2. Choose a unit context (optional). The adjugate entries are pure numbers, but if your matrix came from measurements you can label the entries as metric or US customary, or as money in any of 20 world currencies including the US dollar and the Russian ruble. The label is carried through to your results and printout.
3. Enter the matrix entries. Decimals and negative numbers are both fine.
4. Tick "Compute the inverse" if you also want A⁻¹ = adj(A)/det(A).
5. Read the results: the cofactor matrix, the transpose step that flips the indices, the adjugate itself, the determinant, and the verification identity A·adj(A) = det(A)·I.

Adjugate matrix formula for inverting a matrix

The adjugate is the engine behind one of the classic ways to invert a matrix. The recipe is:

A−1 = adj(A) / det(A)

So the full procedure is:

1. Compute the cofactor matrix cof(A).
2. Transpose it to get the adjugate adj(A).
3. Compute the determinant det(A).
4. Divide every entry of the adjugate by det(A).

This follows directly from the fundamental identity A·adj(A) = det(A)·I: dividing both sides by det(A) shows that adj(A)/det(A) is precisely the inverse.

⚠️ This only works when det(A) ≠ 0. If the determinant is zero the matrix is singular and has no inverse — the adjugate still exists, but you cannot divide by zero.

FAQs

What is the adjugate of a 2×2 matrix?

For A = [a b; c d], the adjugate is [d −b; −c a]. Swap the diagonal entries a and d, then flip the signs of the off-diagonal entries b and c.

Is the adjoint the same as the adjugate?

In this calculator, yes — we use adjoint, adjugate, and classical adjoint interchangeably. Be careful, though: in some texts "adjoint" instead means the conjugate transpose (Hermitian adjoint) A*, which is a completely different object.

What is the relationship between the adjugate and the cofactor matrix?

The adjugate is the transpose of the cofactor matrix: adj(A) = cof(A)^T. Equivalently, adj(A)_ij = C_ji, the cofactor at the flipped index. For symmetric matrices the two coincide.

Can a non-square matrix have an adjugate?

No. The adjugate is built from cofactors, and cofactors are determinants of submatrices — which only exist for square matrices. That's why this calculator only accepts 2×2, 3×3, and 4×4 inputs.

What is A·adj(A)?

It always equals det(A)·I, a scalar multiple of the identity matrix. This holds for every square matrix, singular or not, and is the reason the adjugate gives the inverse when det(A) ≠ 0.

What if the determinant is zero?

The adjugate still exists and is perfectly well defined — only the inverse doesn't. A zero determinant means the matrix is singular: its rows (and columns) are linearly dependent, and it cannot be inverted.