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Show that the matrices $A =\left(\begin{matrix} 5 & 1 \\ -6 & 0 \end{matrix}\right)$ and $B = \left(\begin{matrix} 7 & -5 \\ 4 & -2 \end{matrix}\right) $

are similar over $\mathbb{R}$. In other words, show that there is an invertible matrix $C$ with real coefficients such that $A = C^{-1}BC$.

For this problem I don't know how to show they are similar. I tried to solve for the matrix $C$, by setting up $C = \left(\begin{matrix} a & b \\ c & d\end{matrix}\right)$, such that $CA = BC$. But I feel it's not the easiest way to do this. Please help me figure this out, thank you!

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    $\begingroup$ Do you know about characteristic polynomial, eigenvalues, eigenvectors...? $\endgroup$
    – DonAntonio
    Mar 25, 2016 at 22:23
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    $\begingroup$ First you should check that trace and determinant agree. If so, then just solve your $CA = BC$ thing, it is really not that hard. $\endgroup$
    – Will Jagy
    Mar 25, 2016 at 22:31

1 Answer 1

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Hint: The characteristic polynomial of both matrices is $x^2-5x+6 = (x-3) \cdot (x-2)$. Thus both matrices are diagonalizable with same eigenvalues, so there exist invertible matrices $S, T$ with $S^{-1}AS = T^{-1}BT$.

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