Answer
True
Work Step by Step
Let $A=\begin{bmatrix}
a_{11} & a_{12} \\a_{21} & a_{22}
\end{bmatrix}$ and $B=\begin{bmatrix}
b_{11} & b_{12} \\b_{21} & b_{22}
\end{bmatrix}$ be $2 × 2$ matrices
Then $A+B=\begin{bmatrix}
a_{11} & a_{12} \\a_{21} & a_{22}
\end{bmatrix}+\begin{bmatrix}
b_{11} & b_{12} \\b_{21} & b_{22}
\end{bmatrix}=\begin{bmatrix}
a_{11}+b_{11} & a_{12}+b_{12} \\a_{21}+b_{21} & a_{22}+b_{22}
\end{bmatrix}$
We have $adj(A)=\begin{bmatrix}
a_{22} & -a_{12} \\-a_{21} & a_{11}
\end{bmatrix}\\
adj(B)=\begin{bmatrix}
b_{22} & -b_{12} \\-b_{21} & b_{11}
\end{bmatrix}\\
\rightarrow adj(A)+adj(B)=\begin{bmatrix}
a_{22} & -a_{12} \\-a_{21} & a_{11}
\end{bmatrix}+\begin{bmatrix}
b_{22} & -b_{12} \\-b_{21} & b_{11}
\end{bmatrix}\\
=\begin{bmatrix}
a_{22}+b_{22} & -a_{12}-b_{12} \\-a_{21}-b_{21} & a_{11}+b_{11}
\end{bmatrix}\\
=\begin{bmatrix}
a_{22}+b_{22} & -(a_{12}+b_{12}) \\-(a_{21}+b_{21}) & a_{11}+b_{11}
\end{bmatrix}$
Thus, we can see that $C_{11}=a_{22}+b_{22}\\
C_{12}=-(a_{21}+b_{21})\\
C_{21}=-(a_{22}+b_{22})\\
C_{11}=a_{11}+b_{11}\\
adj(A+B)=\begin{bmatrix}
a_{22}+b_{22} & -(a_{12}+b_{12}) \\-(a_{21}+b_{21}) & a_{11}+b_{11}
\end{bmatrix}$
Consequently, $adj(A)+adj(B)=adj(A+B)$
