Question (and Answer) List from Quantitative Researcher - Part 5. Others

Misc

What are the conditions for an integer matrix to have an inverse such that elements in this inverse are all integers?

Sufficient and necessary condition would be the determinant being $\pm1$.

Easily for necessary, $\mathrm{det}(A^{-1}) = 1/\mathrm{det}(A)$.

For sufficiency, note that by the adjugate matrix, we have $A^{-1} = \mathrm{det}(A)^{-1} \mathrm{adj}(A)$.
Redbook, 红薯哥. 从一个数组中找出第二大的数，最少需要进行多少次比较?

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Redbook, 红薯哥. 已知y对x1和x2分别进行一元线性回归的R方，求y对x1和x2二元回归的R方的取值范围

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Redbook, 维特根斯坦. Given noisy estimates of signed differences $d_{ij}$ between a set of $n$ real numbers ${x_i}$,

find estimators ${\hat x_i}$ that minimize $\sum_{i,j}(d_{ij}-(\hat x_i- \hat x_j))^2$ subject to $\min(\hat x_i) = 0$.

The data is given as a skew-symmetric matrix, $D_{n\times n} = (d_{ij})$, with $d_{ij}\approx x_i - x_j$. Write a program.

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Redbook, 爱跳舞的papaya. 盒子里有 N+1 枚蓝球，N 枚红球，每次摸出蓝球得一元，摸出红球减一元，可以随时停止退出。问这个游戏的期望收益是多少?

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