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Can we find two (different) matrices, $A \in \mathbb{R}^{2\times2}$ and $B \in \mathbb{R}^{2\times2}$ with the same

  • first row (R1): $(5,3)$
  • eigenvalues
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    What do you mean by first line? I ask specifically because your question implies the matrices are $2\times 2$, yet your 'first line' has $3$ entries. – 2017-02-05
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    yes! thank you! now it's correct. – 2017-02-05
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    Is line 1 referring to values in a row, or in a column, or something else? – 2017-02-05
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    I mean 'row', thank you again! (edited) – 2017-02-05

1 Answers 1

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Let $M=\pmatrix{5&3\\a&b}$. Then its characteristic polynomial is

$$p_M(\lambda)=\det\pmatrix{\lambda-5&-3\\-a&\lambda-b}=(\lambda-5)(\lambda-b)-3a=\lambda^2-(5+b)\lambda+(5b-3a)$$

Let $\lambda_1,\lambda_2$ be the roots of $p_M(\lambda)$. Then

\begin{align}\tag{1}\label{eq1}\lambda_1+\lambda_2&=-(5+b)\\ \tag{2}\label{eq2}\lambda_1\cdot\lambda_2&=5b-3a\end{align}

It follows that if $A,B$ have the same eigenvalues, then equation $\eqref{eq1}$ implies that $b$ is the same for both of them. This in turn implies via $\eqref{eq2}$ that $a$ is the same for both of them. Therefore, if $A$ and $B$ have the same eigenvalues, then $A=B$, so the answer to your question is no.

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    Very simple solution. It was easier than I thought, thank you very much! – 2017-02-06