Calculate the classical adjoint (adjugate) and determinant for 2x2 and 3x3 real matrices, and see how cofactors build the final result.
Last updated: June 2026 | By Summa Calculator
In this calculator, adjoint means the classical adjoint, also called the adjugate: the transpose of the cofactor matrix, commonly written adj(A). In other contexts, “adjoint” can instead mean the conjugate transpose, which is a different operation.
The adjoint appears in the matrix inverse formula. If A has a non-zero determinant, then A-1 = adj(A) / det(A). That is one of the main reasons adjoint matrices matter in linear algebra.
Even when the determinant is zero, the adjoint still exists. A matrix with determinant zero is called singular, and singular matrices do not have inverses even though the adjoint can still be computed.
A matrix is usually written with square brackets, and the adjoint is written as adj(A).
To build the adjoint, you first compute the cofactor matrix. A cofactor is the determinant of a smaller matrix formed by removing one row and one column, then applying the alternating sign pattern.
Once the cofactor matrix is complete, transpose it to get the adjugate. Equivalently, the (i, j) entry of adj(A) is the (j, i) cofactor of A.
For a 2x2 matrix, the adjoint is built by swapping the diagonal entries and changing the signs of the off-diagonal entries.
This is the simplest case, and it shows why the adjoint is tied to cofactors rather than just a plain transpose.
For 2x2 matrices, the adjoint is quick to compute because it only requires swapping and sign changes.
For 3x3 matrices, each entry in the cofactor matrix comes from a 2x2 determinant, so the calculation is longer but follows the same idea.
That is why the calculator shows the working steps: it helps you see how each cofactor contributes to the final adjoint.
For every square matrix A, the classical adjoint satisfies A · adj(A) = det(A) · I, where I is the identity matrix.
If det(A) is nonzero, dividing that identity by det(A) gives A-1 = adj(A) / det(A). If det(A) = 0, that inverse formula cannot be used, although adj(A) itself still exists.
This calculator computes the adjugate directly from cofactors; it does not require the matrix to be invertible.
Does the adjoint exist when det(A) = 0?
Yes. The adjoint still exists even if the matrix is singular.
Does “adjoint” always mean adjugate?
No. This calculator uses “adjoint” in the classical-adjoint sense: adj(A) = C^T. In some linear-algebra contexts, adjoint means the conjugate transpose instead.
Can I use 3x3 matrices?
Yes. The calculator supports both 2x2 and 3x3 matrices.
Are decimal results exact?
Not necessarily. The calculator uses JavaScript floating-point arithmetic and displays values to at most eight decimal places; very small or large nonzero values use scientific notation.
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