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

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

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
$A=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right],B=\left[\begin{array}{cc}-1& 1\\ 0& 1\end{array}\right]$
Both are $2×2$ diagonalizable.
Then,
$A+B=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]+\left[\begin{array}{cc}-1& 1\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}0& 2\\ 0& 0\end{array}\right]$

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