Answer to Question #213557 in Linear Algebra for Simphiwe Dlamini

Question

Let T element of L(R³ ) such that -4, 5 and square root 7 are its eigenvalues.

Show that T(x) - 9x = (-4, 5, square 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

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