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4

Part of 1994 Putnam

Problems(2)

Putnam 1994 A4

Source:

7/12/2014
Let AA and BB be 2×22\times 2 matrices with integer entries such that A,A+B,A+2B,A+3B,A, A+B, A+2B, A+3B, and A+4BA+4B are all invertible matrices whose inverses have integer entries. Show that A+5BA+5B is invertible and that its inverse has integer entries.
Putnamalgebrapolynomialquadraticscollege contests
Putnam 1994 B4

Source:

7/12/2014
For n≥1n\ge 1 let dnd_n be the gcd⁡\gcd of the entries of An−I2A^n-\mathcal{I}_2 where A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}  \text{ and }  \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix} Show that lim⁡n→∞dn=∞\lim_{n\to \infty}d_n=\infty.
Putnamlinear algebramatrixlimitinductioncollege contests