# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

##### Exhaustive Sampling and Related Binomial Identities

This  project uses a sampling problem to compute certain probabilities. In the process, certain binomial identities are established.

##### Using Random Tilings to Derive Fibonacci Congruence

This capsule follows the technique of random tilings used in the proof of the closed form for Fibonacci Numbers. By relaxing the condition of probability, the authors are able to obtain congruences relations that involve Fibonacci numbers.

##### Visualizing Leibniz's Rule

The author presents a geometric interpretation of Leibniz's rule for differentiating under the integral sign, and gives an informal visual derivation of the rule.

##### Starting with Two Matrices

The author offers two examples that illustrate important central ideas in introductory linear algebra (independent or dependent vectors; invertible or singular matrices) which may aid students in developing conceptual understanding before any general theory is attempted.

##### A Short Proof of the Two-sidedness of Matrix Inverses

The paper gives a short proof that for any $$n$$ x $$n$$ matrices $$A$$ and $$C$$ over a field of scalars, $$AC = I$$ if and only if $$CA = I$$. The proof relies on familiarity with elementary matrices and the reduced row echelon form.

##### Characteristic Polynomials of Magic Squares

An $$n \times n$$ matrix whose rows, columns, and diagonal all sum to the same number $$m$$ is called magic, and the number $$m$$ is called the magic sum.  If $$A$$ is a magic square matrix, then its magic sum $$m$$ must be an eigenvalue, and hence a characteristic root, of $$A$$.  A main result of this paper shows that the sum of all the characteristic roots of $$A$$ except for $$m$$ must be zero.