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Browse Classroom Capsules and Notes

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Displaying 71 - 80 of 85

An Infinite Series for \(\pi\) with Determinants

The author gives an expression for \(\pi\) involving an infinite sequence of determinants, each representing the area of a triangle.

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...

Wronskian Harmony

A closed form of the Wronskian for \( sin(kx) \) and for \( e^{kx}, k=1,2,\ldots,n \) is obtained. The derivation depends on trigonometric identities and properties of the determinant....

Definitely \(\sim\) Positively the Pits

The author classifies the quadratic forms defined by simple 2 by 2 matrices and illustrates them with corresponding quadratic surfaces.

On the Measure of Solid Angles

The author revisits formulas of measuring solid angles that he could find only in centuries-old literature, and provides modern versions of the proofs.

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...

Root Preserving Transformations of Polynomials

The article answers negatively the question, “Is there a (non-trivial) linear transformation \(T\) from \(P_n\), the vector space of all polynomials of degree at most \(n\), to \(P_n\)...

Tennis with Markov

In the game of tennis, if the probability that player \(A\) wins a point against player \(B\) is a constant value \(p\), then the probability that \(A\) will win a game from deuce is \(p^2/(1 - 2p...

A Direct Proof That Row Rank Equals Column Rank

A row (column) of a matrix is called “extraneous” if it is a linear combination of the other rows (columns).  The author shows that deleting an extraneous row or column of a...

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...

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