Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$

Let $A$ be a $2n\times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1,$ each with probability $1/2.$ Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A.$

Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

Let $a_j$, $b_j$, $c_j$ be integers for $1 \le j \le N$. Assume for each $j$, at least one of $a_j$, $b_j$, $c_j$ is odd. Show that there exists integers $r, s, t$ such that $ra_j+sb_j+tc_j$ is odd for at least $\tfrac{4N}{7}$ values of $j$, $1 \le j \le N$.

Let $n$ be a positive integer. Suppose that $A,B,$ and $M$ are $n\times n$ matrices with real entries such that $AM=MB,$ and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\det(A-MX)=\det(B-XM)$ for every $n\times n$ matrix $X$ with real entries.

Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$

In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are: (1) A player cannot choose an element that has been chosen by either player on any previous turn. (2) A player can only choose an element that commutes with all previously chosen elements. (3) A player who cannot choose an element on his/her turn loses the game. Patniss takes the first turn. Which player has a winning strategy?

For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?

Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$