Determinants
http://www.maa.org/taxonomy/term/42305/0
enFinding 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>On Uniformly Filled Determinants
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/on-uniformly-filled-determinants
<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>Given a square matrix \(U\) and column vectors \( \alpha\) and \( \beta\), the author shows that \( \det(U + \alpha \beta^T) = \det U + \beta^T\)</em> Cof\( (U) \alpha \). <em>This capsule responds to and generalizes a previous classroom capsule.</em></p>
</div></div></div>Determinants of Sums
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/determinants-of-sums
<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>An interesting formula for the determinant of the sum of any two matrices of the same size is presented. The formula can be used to obtain important results about the characteristic polynomial and about the characteristic roots and subdeterminants of the matrices being added.</em></p>
</div></div></div>Determinantal Loci
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/determinantal-loci
<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 characterizes the points \((x, y)\) in the plane for which the determinant of a matrix of a particular form involving \((x, y)\) is \(0\). The matrices of interest have the form \(A+xL+uM\), where \(A\), \(L\), and \(M\) are square matrices, \(L\) and \(M\) are of rank one, and \(L + M\) is of rank two.</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 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>Proof without Words: A 2 x 2 Determinant Is the Area of a Parallelogram
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/proof-without-words-a-2-x-2-determinant-is-the-area-of-a-parallelogram
<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 presents a visual proof that the determinant of a 2 by 2 matrix equals the area of the corresponding parallelogram.</em></p>
</div></div></div>A Polynomial Taking Integer Values
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-polynomial-taking-integer-values
<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 article supplies a short, elementary proof that for integers \(a_1 < a_2 < \cdots < a_n \), the expression \( \prod_{n \geq i > j \geq 1} \frac{a_i - a_j}{i-j} \) is an integer. This previously known result is proved using the Vandermonde determinant. (Please note a typo in the first sentence of the paper where a fraction bar has been omitted.)</em></p>
</div></div></div>Determinants of the Tournaments
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/determinants-of-the-tournaments
<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 a tournament with \(n\) players where each player plays every other player once, and ties are not allowed. An \( n \times n\) tournament matrix \(A\) is constructed where diagonal entries are zero, \(A_{ij} = 1\) if \(i\) beats \(j\), and \(A_{ij}=-1\) if \(j\) beats \(i\). The authors demonstrate that the determinant of a tournament matrix is zero if and only if \(n\) is odd. Additionally, it is shown that the nullspace of a tournament matrix has dimension zero if \(n\) is even and dimension one if \(n\) is odd.</em></p>
</div></div></div>Supermultiplicative Inequalities for the Permanent of Nonnegative Matrices
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/supermultiplicative-inequalities-for-the-permanent-of-nonnegative-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 discusses the relationship of two well-known, apparently unrelated supermultiplicative functions of nonnegative matrices, and shows they are special cases of a more general supermultiplicative function. An application to products of random matrices is sketched.</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>An Infinite Series for \(\pi\) with Determinants
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/an-infinite-series-for-pi-with-determinants
<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 gives an expression for \(\pi\) involving an infinite sequence of determinants, each representing the area of a triangle.</em></p>
</div></div></div>Wronskian Harmony
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/wronskian-harmony
<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>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</em>.</p>
</div></div></div>A Nonstandard Approach to Cramer's Rule
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-nonstandard-approach-to-cramers-rule
<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>Cramer's Rule gives an explicit formulation for the unique solution to a system of \(n\) equations in \(n\) unknowns when the coefficient matrix of the system is invertible. The standard proof is developed using the adjoint matrix. In this capsule, the author uses properties of determinants and general matrix algebra to provide an alternative proof of Cramer's Rule.</em></p>
</div></div></div>An Alternate Proof of Cramer's Rule
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/an-alternate-proof-of-cramers-rule
<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 provides a short proof of Cramer’s rule that avoids using the adjoint of a matrix.</em></p>
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