# Browse Classroom Capsules and Notes

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Displaying 41 - 50 of 85

There are three somewhat distinct topics in this article: the condition for linear functions to commute, a linear function as a transformation of the number line, and linear difference...

This article provides a non-group theory approach to finding the number of two by two matrices over $Z/pZ$ that have both eigenvalues in the same field.  The strategy is to use the...

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...

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...

If noise in data transmission produces a not quite orthogonal matrix that is known to be orthogonal, how does one find the "nearest" orthogonal matrix?  This capsule recasts the...

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...

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

Given any 3 by 3 rotation matrix $A$ (i.e. orthogonal with determinant $1$ and an arbitrary vector $x$, the vector $Ax +A^{T}x+[1−$tr$(A)]x$ lies on the axis of rotation. The...