Question: If a matrix is small, its inverse is large. If a number \(a\) is small, its inverse \(1 / a\) (assuming \(a \neq 0\) ) is large. In this exercise you will explore a matrix analog of this idea. Suppose the \(n \times n\) matrix \(A\) is invertible. Show that \(\left\|A^{-1}\right\| \geq \sqrt{n} /\|A\|\). This implies that if a matrix is small, its inverse is large. Hint. You can use the inequality \(\|A B\| \leq\|A\|\|B\|\), which holds for any matrices for which the product makes sense.

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