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  4. The Eigen values of the matrix [x-a 0 0 x-b x-c 0 x-d x-e x-f] are equal ∀ x, then

Q. The Eigen values of the matrix $\begin{bmatrix}x-a & 0 & 0 \\ x-b & x-c & 0 \\ x-d & x-e & x-f\end{bmatrix}$ are equal $\forall x$, then

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Solution:

Since the matrix is triangular the diagonal elements are the Eigen values.
By the given condition, $x-a=x-c=x-f $
$ \therefore a=c=f$.

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