The Banach Lemma: Let $B$ be an n x n matrix . If in some induced matrix norm $ \|B\| < 1$, then $ I + B $ is invertible and $\|(I + B)\|^{-1} ≤ \frac1{(1− \| B \|)}.$
Question: Consider the n by n tridiagonal matrix $T_n$ = $diag[1, 3, 1]$ . For example, $$T_4 =\pmatrix{3 & 1 & 0& 0\cr 1&3 &1& 0\cr 0&1 &3 &1\cr 0 & 0& 1& 3} $$ Use the Banach Lemma to show that $T_n$ is invertible for all positive integers n.
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^ This is an assignment question I have, but I don't understand how I am supposed to answer it if when, clearly, $\|T_4\|_\infty = \|T_4\|_1 = 5 $ which does not satisfy the condition since n is 4 (positive) and $ \|B\|$ should be $ < 1$.
Is there something that I don't understand correctly? The prof has given this assignment for a few years now, I am thinking it is highly improbable that there there is a mistake in the question.
Any help is appreciated.
