# Browse Classroom Capsules and Notes

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Displaying 71 - 80 of 85

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

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...

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....

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

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

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...

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 $P_n$...

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 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...

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...