How this calculator works
The adjugate is built from cofactors: replace each entry by the signed determinant of its minor, then transpose the result. It gives a compact formula for the inverse, A inverse equals adj(A) divided by det(A), which is useful for small matrices. The identity A adj(A) = det(A) I holds for every square matrix. When the determinant is zero the adjugate still exists but no inverse does.
Formula
adj(A) = C^T, where Cij = (-1)^(i+j) det(Mij). A adj(A) = det(A) I.
Worked example
For the matrix with rows 1, 2 and 3, 4, the cofactor matrix is 4, -3; -2, 1, so the adjugate is 4, -2; -3, 1. The determinant is -2, and A times the adjugate is -2 times the identity.
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Frequently asked questions
Is the adjoint the same as the conjugate transpose?
Not in this context. In linear algebra the adjoint or adjugate of a matrix here means the transposed cofactor matrix. In complex analysis, adjoint can mean conjugate transpose.
What happens when the matrix is singular?
The adjugate is still defined, but dividing by a zero determinant is impossible, so the matrix has no inverse.
Can I use the adjugate for a 4 by 4 matrix?
Yes. The page supports 2 by 2, 3 by 3 and 4 by 4 matrices. A 4 by 4 adjugate uses sixteen 3 by 3 minor determinants.