Calculate the adjoint and determinant for 2x2 and 3x3 matrices, and see how cofactors build the final result.
Last updated: June 2026 | By Patchworkr Team
The adjoint of a matrix is the transpose of its cofactor matrix. It is commonly written as adj(A).
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, you transpose it to get the adjoint. That is the step that turns the row-and-column cofactors into adj(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.
Does the adjoint exist when det(A) = 0?
Yes. The adjoint still exists even if the matrix is singular.
Is this the same as the transpose?
No. The adjoint is the transpose of the cofactor matrix, not the simple transpose.
Can I use 3x3 matrices?
Yes. The calculator supports both 2x2 and 3x3 matrices.
Why must all entries be valid?
The adjoint requires exact arithmetic from every matrix entry.
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