Answer to Question #213557 in Linear Algebra for Simphiwe Dlamini

Question #213557

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).


1
Expert's answer
2022-02-01T13:12:20-0500

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 xR3x\isin R^3 such that

(T9I)x=Tx9x=(4,5,7)(T − 9I)x = T x − 9x = (4, −5, \sqrt7)

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