Adjoint

Suppose $A=(a_{ij})_{n\times{n}}$. $$

$$\text{adj}A = (A_{ij})_{n\times{n}}^T$$

Where $A_{ij}$ is the co-factor of $a_{ij}$.

Properties

Suppose $A$ is a $n\times n$ matrix.

• $\text{adj}(I)=I$
• $\text{adj}(cA)=c^{n-1}\text{adj}(A)$
• $\text{adj}(A^T)=(\text{adj}(A))^T$
• $\text{adj}(A)\,A = A\,\text{adj}(A) = \lvert A \rvert I$
• $A\,(\text{adj}A) = (\text{adj}A)\,A = \lvert{A}\rvert I$

Note

For a $2\times 2$ matrix, $\text{adj}(\text{adj}(A))=A$.