Question

Let T€L(R) such that - 4,5 and root 7 are its eigenvalues. Show that T(x) - 9x=(- 4,5, root 7)

Answer 1

Since T has at most 3 distinct eigenvalues , the hypothesis imply that 9 is not an eigenvalue of T.

Thus T − 9I is surjective. In particular, there exists $x\isin R^3$ such that

$(T − 9I)x = T x − 9x = (4, −5, \sqrt7)$