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Find an example of matrices, $A$ and $B$, with $AB=BA$ and for which $\lambda$ is an eigenvalue of $A$, $\mu$ an eigenvalue of $B$, but $\lambda+\mu$ is not an eigenvalue of $A+B$, and $\lambda \mu$ not an eigenvalue of $AB$. Can anyone please provide an example of two such matrices?matrices should not be triangular and diagonal

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You have a low answer-acceptance rate. Please read about accepting answers here and here. –  Git Gud Mar 22 at 13:31
    
Pick for A and B two matrices that are really easy to calculate with that satisfy the conditions. Which ones did you pick? Do they work? If not, why not? –  I like Serena Mar 22 at 13:53
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"low"?? Not even a single accepted Question/Answer.. I am not sure if any body would wish to help you if you do not react properly for users who spared their time to help you... –  Praphulla Koushik Mar 22 at 14:23
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2 Answers 2

up vote 2 down vote accepted

Ok trying again. Take $$A = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 0\end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 0 & 1\\ 0 & 2 & 0\\ 1 & 0 & 1\end{bmatrix}\,.$$

These matrices commute, neither is diagonal, and neither is triangular.

Eigenvalues of $A$: $-1, 1, 0$.

Eigenvalues of $B$: $2, 2, 0$.

Eigenvalues of $A+B$: $3,2,-1$.

Eigenvalues of $AB$: $2,0,0$.

So take $\lambda = -1$ and $\mu = 2$.

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$A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has eigenvalue 1, $B=\begin{pmatrix}0&2\\2&0\end{pmatrix}$ has eigenvalue -2

$A+B=\begin{pmatrix}0&3\\3&0\end{pmatrix}$ does not have eigenvalue $1-2=-1$

$AB=\begin{pmatrix}2&0\\0&2\end{pmatrix}$ does not have eigenvalue $1\cdot-2=-2$

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A better answer than mine, obviously! –  Jason Zimba Mar 22 at 15:37
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Nice one.${{}}$ –  Git Gud Mar 22 at 15:53
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