Determinants
https://www.maa.org/taxonomy/term/42305/all
enA Geometrical Approach to Cramer's Rule
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-geometrical-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>Motivated by the interpretation of a determinant of a \(2 \times 2\) matrix as the area of a parallelogram, the author derives Cramer's rule geometrically.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95056 at https://www.maa.orgFinding a Determinant and Inverse Matrix by Bordering
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95091 at https://www.maa.orgThe Existence of Multiplicative Inverses
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95149 at https://www.maa.orgThe Square Roots of \(2 \times 2\) Matrices
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95303 at https://www.maa.orgProof without Words: A 2 x 2 Determinant Is the Area of a Parallelogram
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95382 at https://www.maa.orgA Polynomial Taking Integer Values
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95399 at https://www.maa.orgDeterminants of the Tournaments
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95402 at https://www.maa.orgSupermultiplicative Inequalities for the Permanent of Nonnegative Matrices
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95449 at https://www.maa.orgA Transfer Device for Matrix Theorems
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95491 at https://www.maa.orgAn Infinite Series for \(\pi\) with Determinants
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95601 at https://www.maa.orgWronskian Harmony
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95655 at https://www.maa.orgA Nonstandard Approach to Cramer's Rule
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95773 at https://www.maa.orgAn Alternate Proof of Cramer's Rule
https://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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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95777 at https://www.maa.orgDeterminantal Loci
https://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>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin94792 at https://www.maa.orgApropos Predetermined Determinants
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/apropos-predetermined-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 illustrates how certain determinants can be used to motivate students. The determinants in question have terms in artithmetic progressions.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
newton_admin95783 at https://www.maa.org