Denote by $C_{n, m}$ the $(n, m)$th
D5941: Real square matrix cofactor of $A$. Since we have
\begin{equation}
\begin{split}
A \text{Adj}(A)
& =
\begin{bmatrix}
A_{1, 1} & \cdots & A_{1, N} \\
\vdots & \ddots & \vdots \\
A_{N, 1} & \cdots & A_{N, N}
\end{bmatrix}
\begin{bmatrix}
C_{1, 1} & \cdots & C_{N, 1} \\
\vdots & \ddots & \vdots \\
C_{1, N} & \cdots & C_{N, N}
\end{bmatrix}
\\
& =
\begin{bmatrix}
\sum_{n = 1}^N A_{1, n} C_{1, n} & \cdots & \sum_{n = 1}^N A_{1, n} C_{N, n} \\
\vdots & \ddots & \vdots \\
\sum_{n = 1}^N A_{N, n} C_{1, n} & \cdots & \sum_{n = 1}^N A_{N, n} C_{N, n}
\end{bmatrix}
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\text{Adj}(A) A
& =
\begin{bmatrix}
C_{1, 1} & \cdots & C_{N, 1} \\
\vdots & \ddots & \vdots \\
C_{1, N} & \cdots & C_{N, N}
\end{bmatrix}
\begin{bmatrix}
A_{1, 1} & \cdots & A_{1, N} \\
\vdots & \ddots & \vdots \\
A_{N, 1} & \cdots & A_{N, N}
\end{bmatrix}
\\
& =
\begin{bmatrix}
\sum_{n = 1}^N C_{n, 1} A_{n, 1} & \cdots & \sum_{n = 1}^N C_{n, 1} A_{n, N} \\
\vdots & \ddots & \vdots \\
\sum_{n = 1}^N C_{n, N} A_{n, 1} & \cdots & \sum_{n = 1}^N C_{n, N} A_{n, N}
\end{bmatrix}
\end{split}
\end{equation}
then this result is a consequence of
R5064: Cofactor partition for a real square matrix. $\square$
