Matrix Algebra
https://maa.org/taxonomy/term/41377/all
enPascal Matrices
https://maa.org/programs/maa-awards/writing-awards/pascal-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 class="MsoNormal"><b>Year of Award</b>: 2005</p><p class="MsoNormal"><b>Publication Information:</b> <i>The American Mathematical Monthly</i>, vol. 111, no. 3, March 2004, pp. 361-385.</p><p class="MsoNormal"><strong>Summary:</strong> This paper proves an interesting factorization theorem for a family of square matrices built from Pascal's triangle in a natural way.</p><p class="MsoNormal"><a title="Read the Article:" href="/sites/default/files/pdf/upload_library/22/Ford/Edelman189-197.pdf">Read the Article:</a> </p></div></div></div>Permanents
https://maa.org/programs/maa-awards/writing-awards/permanents
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal"><strong>Year of Award</strong>: 1966</p>
<p class="MsoNormal"><strong>Publication Information:</strong> <em>The American Mathematical Monthly</em>, vol. 72, 1965, pp. 577-591</p></div></div></div>A Disorienting Look at Euler's Theorem on the Axis of a Rotation
https://maa.org/programs/maa-awards/writing-awards/a-disorienting-look-at-eulers-theorem-on-the-axis-of-a-rotation
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span style="font-family: verdana,geneva;"><span style="font-size: small;"><span style="font-family: andale mono,times;"><span style="font-size: xx-small;"><span style="font-family: verdana,geneva;"><span style="font-size: small;"><strong>Award:</strong> Lester R. Ford</span></span></span></span></span></span></p></div></div></div>Computing the Determinant Through the Looking Glass
https://maa.org/press/periodicals/convergence/computing-the-determinant-through-the-looking-glass
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Project in which students learn a simple and efficient way to compute determinants from a paper of Charles Dodgson (Lewis Carroll)</p>
</div></div></div>The Many Names of (7,3,1)
https://maa.org/programs/maa-awards/writing-awards/the-many-names-of-731
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><b>Award:</b> Carl B. Allendoerfer</p>
<p><b>Year of Award:</b> 2003</p>
<p><b>Publication Information:</b> <i>Mathematics Magazine</i>, Vol. 75(2002), pp. 83-94</p>
<p><b>Summary:</b> One object that is a difference set, a block design, a Steiner triple system, a finite projective plane, a complete set of orthogonal Latin Squares, a doubly regular round-robin tournament, a skew-Hadamard matrix, and a graph consisting of seven mutually adjacent hexagons drawn on the torus.</p></div></div></div>Gaussian Elimination in Integer Arithmetic: An Application of the \(L\)-\(U\) Factorization
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/gaussian-elimination-in-integer-arithmetic-an-application-of-the-l-u-factorization
<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 L-U factorization, the author generates examples of matrices for which Gaussian elimination process can be done in integer arithmetic, including examples of matrices that are invertible over the integers.</em></p>
</div></div></div>Notational Collisions
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/notational-collisions
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>This capsule points out several potential confusions in commonly used linear algebra notation.</p>
</div></div></div>The Square Roots of \(2 \times 2\) Matrices
https://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>Matrices as Sums of Invertible Matrices
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrices-as-sums-of-invertible-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 any square matrix over a field is the sum of two invertible matrices, and that the decomposition is unique only if the matrix is nonzero and of size 2x2 with entries in the field of two elements.</em></p>
</div></div></div>A Transfer Device for Matrix Theorems
https://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>Polynomial Translation Groups
https://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>Characteristic Polynomials of Magic Squares
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/characteristic-polynomials-of-magic-squares
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>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, then its magic sum \(m\) must be an eigenvalue, and hence a characteristic root, of \(A\). A main result of this paper shows that the sum of all the characteristic roots of \(A\) except for \(m\) must be zero.</p>
</div></div></div>A Short Proof of the Two-sidedness of Matrix Inverses
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-short-proof-of-the-two-sidedness-of-matrix-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>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 \(CA = I\).</em> <em>The proof relies on familiarity with elementary matrices and the reduced row echelon form.</em></p>
</div></div></div>A Direct Proof That Row Rank Equals Column Rank
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-direct-proof-that-row-rank-equals-column-rank
<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 row (column) of a matrix is called “extraneous” if it is a linear combination of the other rows (columns). The author shows that deleting an extraneous row or column of a matrix does not affect the row rank or column rank of a matrix. This fact establishes the theorem in the title.</em></p>
</div></div></div>Matrix Inverse Applet
https://maa.org/programs/faculty-and-departments/course-communities/matrix-inverse-applet