If A and B are both n times n matrices (of the same size), then det(A+B)=det(A)+det(B) True or False?

If A and B are both $n×n$ matrices (of the same size), then
det(A+B)=det(A)+det(B)
True or False?
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stuth1
Step 1
In general, the given identity det(A+B)=det(A)+det(B) does not hold true. To prove the given identity is wrong, it is enough to give a counter example. Consider the matrices
Evaluate det(A) as follows.
$det\left(A\right)=|\begin{array}{cc}1& 0\\ 0& 0\end{array}|=0$
Evaluate det(B) as follows.
$det\left(B\right)=|\begin{array}{cc}0& 0\\ 0& 1\end{array}|=0$
Step 2
Evaluate det(A)+det(B) as follows.
det(A)+det(B)=0+0=0
Thus, det(A)+det(B)=0
Evaluate A+B as follows.
$A+B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]=$
$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Evaluate det(A+B) as follows.
$det\left(A+B\right)=|\begin{array}{cc}1& 0\\ 0& 1\end{array}|$
=1
Thus, det(A+B)=1
Hence it is proved that $det\left(A\right)+det\left(B\right)\ne det\left(A+B\right)$
Therefore, the given statement is FALSE.
Jeffrey Jordon