Let $A \in \mathbb{R}^{2 \times 2}$ be a
D6160: Real square matrix such that
| (i) |
\begin{equation}
A
=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\end{equation}
|
The cofactors of $A$ are
\begin{equation}
\begin{split}
C_{1, 1} & = (-1)^{1 + 1} \text{Det} \begin{bmatrix} d \end{bmatrix} = d \\
C_{1, 2} & = (-1)^{1 + 2} \text{Det} \begin{bmatrix} c \end{bmatrix} = -c \\
C_{2, 1} & = (-1)^{2 + 1} \text{Det} \begin{bmatrix} b \end{bmatrix} = -b \\
C_{2, 2} & = (-1)^{2 + 2} \text{Det} \begin{bmatrix} a \end{bmatrix} = a \\
\end{split}
\end{equation}
Thus, the cofactor matrix is
\begin{equation}
\text{Cof} A
=
\begin{bmatrix}
C_{1, 1} & C_{1, 2} \\
C_{2, 1} & C_{2, 2}
\end{bmatrix}
=
\begin{bmatrix}
d & - c \\
- b & a
\end{bmatrix}
\end{equation}
$\square$
