If A and B are n times n diagonalizable matrices , then A+B is also diagonalizable. True or False?

permaneceerc 2020-12-29 Answered
If A and B are n×n diagonalizable matrices , then A+B is also diagonalizable.
True or False?
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Expert Answer

liannemdh
Answered 2020-12-30 Author has 106 answers

Step 1
Given statement,
If A and B are n×n diagonalizable matrices, then A+B is also diagonalizable.
Step 2
This statement is false.
For example, consider
A=[1101],B=[1101]
Both are 2×2 diagonalizable.
Then,
A+B=[1101]+[1101]
=[0200]

Step 3
Since, A+B has eigenvalues 0, 0 and only one linear independent eigenvector corresponding to eigenvalue 0.
So,
A+B is not diagonalizable.

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Jeffrey Jordon
Answered 2022-01-23 Author has 2047 answers

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