3 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
5 & - 1 & 7
0 & 6 & 0
7 & 7 & 5
\end{array} \right) .$$
- Find the eigenvalues of \(\mathbf { A }\).
- Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
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The diagram shows the curve with equation \(\mathrm { y } = \ln \mathrm { x }\) for \(x \geqslant 1\), together with a set of ( \(N - 1\) ) rectangles of unit width. - By considering the sum of the areas of these rectangles, show that
$$\ln N ! > N \ln N - N + 1 .$$
- Use a similar method to find, in terms of \(N\), an upper bound for \(\operatorname { In } N\) !.