Consider the matrices $A$ and $T$ given by $$A=\begin{bmatrix} 3 &\alpha &\beta\\-1 &7 &-1\\0 &0 &5 \end{bmatrix} , T=\begin{bmatrix} 1 &0 &0\\0 &1/2 &0\\0 &0 &1/4 \end{bmatrix}$$ with $|\alpha|,|\beta| \leq 1.$ Use the similarity trasform $T^{-1}AT$ to show that the matrix $A$ has at least two distinct eigenvalues (hint: Gershgorin's theorem).
I can find the matrix of $T^{-1}AT$, then I found two of the 3 Gershgorin circles of this matrix are disjoint, so I can say there are at least two distinct eigenvalues of $T^{-1}AT$ and hence so is $A$? (since they have the same eigenvalues)
Not sure whether this is the right use of the similarity transform and Gershgorin's theorem. Thanks.