Work out adjoint of a matrix instantly with clear inputs, formula shown and shareable results.
The adjugate is the transpose of the cofactor matrix. It satisfies A·adj(A) = det(A)·I, which is exactly why the inverse is adj(A) divided by the determinant.
Adjugate identity
A · adj(A) = det(A) · I, so A⁻¹ = adj(A) / det(A)
The determinant is 1, so the adjugate and the inverse coincide: [[−24, 18, 5], [20, −15, −4], [−5, 4, 1]].
Yes, always. It is the inverse that fails to exist when the determinant is zero.