Adjugate matrix

2020 Mathematics Subject Classification: Primary: 15A15 [MSN][ZBL]

adjoint matrix

The signed transposed matrix of cofactors for a given square matrix $A$. The $(i,j)$ entry of $\mathrm{adj}\,A$ is $$ \mathrm{adj}\, A_{ij} = (-1)^{i+j} \det A(j,i) $$ where $A(j,i)$ is the minor formed by deleting the row and column through the matrix entry $A_{ji}$.

The expansion of the determinant in cofactors is expressed as $$ A . \mathrm{adj}\, A = \mathrm{adj}\, A . A = (\det A) I $$ where $I$ is the identity matrix.

Some texts do not include the transposition in their definition. The term adjoint matrix is also used, but more commonly refers to the conjugate transpose of a matrix, see Adjoint matrix.

References

• A.C. Aitken, "Determinants and matrices", Oliver and Boyd (1939) Zbl 65.1111.05 Zbl 0022.10005