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

permaneceerc

permaneceerc

Answered question

2020-12-29

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

Answer & Explanation

liannemdh

liannemdh

Skilled2020-12-30Added 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.

Jeffrey Jordon

Jeffrey Jordon

Expert2022-01-23Added 2605 answers

Answer is given below (on video)

Jeffrey Jordon

Jeffrey Jordon

Expert2022-08-23Added 2605 answers

Answer is given below (on video)

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