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?

beljuA 2020-12-03 Answered
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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Expert Answer

stuth1
Answered 2020-12-04 Author has 97 answers
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 A=[1000] and B=[0001]
Evaluate det(A) as follows.
det(A)=|1000|=0
Evaluate det(B) as follows.
det(B)=|0001|=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=[1000]+[0001]=
=[1001]
Evaluate det(A+B) as follows.
det(A+B)=|1001|
=1
Thus, det(A+B)=1
Hence it is proved that det(A)+det(B)det(A+B)
Therefore, the given statement is FALSE.
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Jeffrey Jordon
Answered 2022-01-29 Author has 2070 answers

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