Adjugate Matrix

The adjugate matrix is the transpose of the matrix of cofactors of $\mathbf{A}$. A cofactor is defined as

$$C_{ij}=(-1{)}^{i+j}|M_{ij}|$$

where $|M_{ij}|$ is the determinant of the minor matrix for $A_{ij}$ .

The adjugate matrix is defined as

$$\text{adj}\mathbf{A}={\mathbf{C}}^{⊺}$$

Applications §

Matrix Inverse §

The adjugate can be used to solve for the inverse of a matrix.

Determinant Diagonal §

$$\mathbf{A}(\text{adj}\mathbf{A})=\left(\begin{array}{cccc}\det \mathbf{A}& 0& 0& \dots \\ 0& \det \mathbf{A}& 0& \dots \\ 0& 0& \det \mathbf{A}& \dots \\ 0& 0& 0& \det \mathbf{A}\end{array}\right)$$

Proof §

We can see that

$$\mathbf{A}(\text{adj}\mathbf{A}{)}_{ij}=a_{i1}{C}_{j1}+a_{2i}{C}_{}$$

The elements along the diagonal are equal to the determinant by definition. Take ${\mathbf{A}}_{1j}$ for example:

$$\mathbf{A}(\text{adj}\mathbf{A}{)}_{1j}={\mathbf{A}}_{11}\cdot |{\mathbf{M}}_{11}|+{\mathbf{A}}_{12}\cdot |{\mathbf{M}}_{12}|\dots$$