Matrix Algebra
http://www.maa.org/taxonomy/term/42313/0
enApproaches to the Formula for the \(n\)th Fibonacci Number
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/approaches-to-the-formula-for-the-nth-fibonacci-number
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>In this capsule proofs of the equivalence of the two definitions of the Fibonacci numbers are discussed. This helps the undergraduate view mathematics as a unified whole with a variety of techniques.</em></p>
</div></div></div>Matrix Patterns and Undetermined Coefficients
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrix-patterns-and-undetermined-coefficients
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The Method of Undetermined Coefficients is approached by using linear operators and making use of patterns associated with matrix multiplication. The authors discuss several pedagogical advantages to this approach.</em></p>
</div></div></div>The Gunfight at the OK Corral
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-gunfight-at-the-ok-corral
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article uses a generalization of a three-way gunfight to motivate the construction and solution to a first order linear system of difference equations. The method of undetermined coefficients is used to develop a general solution to the dynamical system. Probabilities of the system converging to each final (absorbing) state are found. According to the author, many mathematical models can be approached from the point of view of discrete dynamical systems.</em></p>
</div></div></div>Linear Algebra in the Financial World
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/linear-algebra-in-the-financial-world
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Linear algebra is used to study financial trading strategies and expectations. Financial conditions are examined via matrix equations, using rank, column space, and null space arguments.</em></p>
</div></div></div>A Diagonal Perspective on Matrices
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-diagonal-perspective-on-matrices
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The authors present real matrices from a diagonal perspective, to supplement the usual row/column perspective and to offer contexts in which this is a useful mode.</em></p>
</div></div></div>Finding a Determinant and Inverse Matrix by Bordering
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/finding-a-determinant-and-inverse-matrix-by-bordering
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The authors present a procedure for finding the determinant and inverse of a special class of matrices. The strategy adds borders to the original matrix, and makes use of row operations and determinant rules.</em></p>
</div></div></div>When is Rank Additive?
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/when-is-rank-additive
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This capsule presents necessary and sufficient conditions for the matrix rank of a sum to be the sum of the ranks. The crux of the argument uses the fact that the rank of a matrix is the size of its largest invertible submatrix.</em></p>
</div></div></div>The Existence of Multiplicative Inverses
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-existence-of-multiplicative-inverses
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Using only basic ideas from linear algebra and number theory, the authors show that if \(c\) is square-free, the ring \(Q [\sqrt[n]{c}] \) is a field. An arbitrary nonzero element of the ring is associated with a system of equations, and divisibility arguments are used to show that a matrix of coefficients from the system must have a nonzero determinant, eventually leading to the result that the original element of the ring has an inverse. </em></p>
</div></div></div>The Arithmetic of Algebraic Numbers: An Elementary Approach
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-arithmetic-of-algebraic-numbers-an-elementary-approach
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>If \(r\) and \(s\) are algebraic numbers, then \(r + s\), \(rs\), and \(r/s\) are also algebraic. The proof provided in this capsule uses the ideas of characteristic polynomials, eigenvalues, and eigenvectors.</em></p>
</div></div></div>The Square Roots of \(2 \times 2\) Matrices
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-square-roots-of-2-times-2-matrices
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The Cayley-Hamilton theorem may be used to determine explicit formulae for all the square roots of \(2 \times 2\) matrices. </em>These formulae indicate exactly when a \(2 \times 2\) matrix has square roots, and the number of such roots.</p>
</div></div></div>Finite Groups of 2 x 2 Integer Matrices
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/finite-groups-of-2-x-2-integer-matrices
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The author shows that a finite group \(G\) can be represented as a group of invertible \( 2 \times 2\) integer matrices if and only if \(G\) is isomorphic to a subgroup of the dihedral groups \(D_4\) or \(D_6\). Results are obtained by studying the relation between \(GL(2,\mathbf{Z})\) and \(SL(2,3)\).</em></p>
</div></div></div>A Transfer Device for Matrix Theorems
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-transfer-device-for-matrix-theorems
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>The author presents a method to transfer matrix identities over the real numbers to matrix identities over an arbitrary commutative ring. Several examples are given, including \(\det(AB)= \det(A) \det(B) \), the Cayley-Hamilton Theorem, and identities involving adjoint matrices.</p>
</div></div></div>Matrices, Continued Fractions, and Some Early History of Iteration Theory
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrices-continued-fractions-and-some-early-history-of-iteration-theory
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Continued fractions of the form \( \frac{1}{1 + \frac{c}{1 + \frac{c}{ 1 +\ddots}}} \) are analyzed using linear algebra and iteration theory. The continued fractions of interest are closely related to a class of \(2 \times 2\) matrices, and the eigenvalues and eigenvectors of those matrices are investigated to determine when the corresponding continued fractions converge. Historical references are included.</em></p>
</div></div></div>A Matrix Proof of Newton's Identities
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-matrix-proof-of-newtons-identities
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Newton’s identities relate the coefficients of a polynomial to sums of powers of its roots. The author uses the Cayley-Hamilton theorem and properties of the trace of a matrix to derive Newton’s identities.</em></p>
</div></div></div>Polynomial Translation Groups
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/polynomial-translation-groups
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>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 \(p(x+r)\), where \(r\) is a real number. The group structure of the matrices \(M(r)\) under multiplication then gives rise to various combinatorial identities.</em></p>
</div></div></div>