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Browse Classroom Capsules and Notes

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Displaying 1 - 10 of 85

Cramer's Rule is due to Cramer

The author reviews the evidence of authorship of Cramer`s Rule and comes to the conclusion stated in the title of the paper.

Fibonacci Identities via the Determinant Sum Property

This capsule uses the determinants of matrices to study Fibonacci numbers. Specifically, the sum property of the determinant is used to derive identities between Fibonacci numbers.

The Existence of Multiplicative Inverse

This capsule discusses a way to show each non-zero element of certain rings have a multiplicative inverse. The approach is to set up a system of linear equations and the solution is the...

An Easy Solution to 'Mini Lights Out'

Using basic linear algebra, the authors obtain a complete, easy to compute, winning strategy for the \(4 \times 4\) mini `Lights Out` game.

Magic "Squares" Indeed!

A real matrix is called square-palindromic if, for every base \(b\), the sum of the squares of its rows, columns, and four sets of diagonals (as described in the article) are unchanged when...

Correcting Cramer's Rule

The author clarifies the wording of Cramer's rule, sidestepping a common misconception.  The Kronecker-Capelli theorem is introduced to help see Cramer's rule in a more complete...

An Application of Sylvester's Rank Inequality

Using two well known criteria for the diagonalizability of a square matrix plus an extended form of Sylvester's Rank Inequality, the author presents a new condition for the diagonalization...

An Elementary Treatment of General Inner Products

A typical first course on linear algebra is usually restricted to vector spaces over the real numbers and the usual positive-definite inner product.  Hence, the proof that \(\dim(\mathcal{S...

An Intuitive Proof of the Singular Value Decomposition of a Matrix

Using a simple trigonometric limit, the author provides an intuitive geometric proof of the Singular Value Decomposition of an arbitrary matrix.

A Tricky Linear Algebra Example

In this article a classroom "trick" involving square arrangements of natural numbers is used to motivate a discussion of a special class of matrices. In particular, a basis is...

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