# Browse Classroom Capsules and Notes

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Displaying 51 - 60 of 85

The article supplies a short, elementary proof that for integers $a_1 < a_2 < \cdots < a_n$, the expression $\prod_{n \geq i > j \geq 1} \frac{a_i - a_j}{i-j}$ is an integer...

Consider a tournament with $n$ players where each player plays every other player once, and ties are not allowed.  An $n \times n$ tournament matrix $A$ is constructed where...

The fact that the limit of the $n$-th power of a $2\times 2$ matrix $A$ tends to $0$ if  $\det A < 1$ and $\mid 1 + \det(A) \mid > \mid$ tr$(A) \mid$ is used...

The author shows that a finite group $G$ can be represented as a group of invertible $2 \times 2$ integer matrices if and only if $G$ is isomorphic to a subgroup of the dihedral groups...

The author shows that the volume of an ellipsoid can be determined by three parallel slices.

Given symmetric unit vectors $u_i$, conditions on real numbers $x_i$ are considered to be able to conclude that $\sum x_i u_i \Rightarrow x_i = 0$ for all $i$. Different kinds of...

The author discusses the relationship of two well-known, apparently unrelated supermultiplicative functions of nonnegative matrices, and shows they are special cases of a more general...

The author derives formulas for the coefficients of the characteristic polynomial of a square matrix in terms of the partial derivatives of its determinant, which is viewed as a function of...

Suppose we have several alternatives that we wish to rank.  For example, we may wish to rank five teachers according to their teaching excellence.  The author constructs a positive...

Consider $n$ vectors issuing from the origin in $n$-dimensional space.  The author shows that the statement “any set of $n$ vectors in $n$-space, no two of which meet at greater...