# Browse Classroom Capsules and Notes

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Displaying 31 - 40 of 85

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 $P_n$...

Using mathematical induction, the author provides a simple proof that the reduced row echelon form of a matrix is unique.

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...

Consider the vector space of polynomials of degree less than $n$, and a polynomial $p(x)$ in this space. The author describes the matrix $M(r)$ that maps the polynomial $p(x)$ to $p(... The Cayley-Hamilton theorem may be used to determine explicit formulae for all the square roots of \(2 \times 2$ matrices. These formulae indicate exactly when a $2 \times 2$...