Adjoint Matrices

Definition: Let $A$ be an $n \times n$ matrix, and let $C_{ij}$ be the cofactor entry of $a_{ij}$, and $\mathrm{cof} (A) = \begin{bmatrix} C_{11} & C_{12} & \cdots & C_{1n}\\ C_{21} & C_{22} & \cdots & C_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ C_{n1} & C_{n2} & \cdots & C_{nn} \end{bmatrix}$ be the matrix of cofactors from $A$. Then the Adjoint Matrix of $A$ denoted $\mathrm{adj} (A) = [\mathrm{cof} (A)]^T$, namely $\mathrm{adj}(A) = \begin{bmatrix} C_{11} & C_{21} & \cdots & C_{n1}\\ C_{12} & C_{22} & \cdots & C_{n2}\\ \vdots & \vdots & \ddots & \vdots\\ C_{1n} & C_{2n} & \cdots & C_{nn} \end{bmatrix}$.

For example, consider the following $3 \times 3$ matrix $\begin{bmatrix}3 & 1 & 3\\ 2 & 3 & 1\\ 1 & 2 & 3\end{bmatrix}$. Let's first calculate all of the cofactors for this matrix, that is:

(1)

\begin{align} C_{11} = 7 \\ C_{12} = -5 \\ C_{13} = 1 \\ C_{21} = 3 \\ C_{22} = 6 \\ C_{23} = -5 \\ C_{31} = -8 \\ C_{32} = 3 \\ C_{33} = 7 \\ \end{align}

Therefore we can construct a matrix of cofactors of $A$, that is:

(2)

\begin{align} \mathrm{cof}(A) = \begin{bmatrix} 7 & -5 & 1\\ 3 & 6 & -5\\ -8 & 3 & 7 \end{bmatrix} \end{align}

If we now take the transpose of this matrix, we thus get our adjoint of $A$, that is:

(3)

\begin{align} \mathrm{adj}(A) = \begin{bmatrix} 7 & 3 & -8\\ -5 & 6 & 3\\ 1 & -5 & 7 \end{bmatrix} \end{align}