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Class 12 Mathematics Topic Wise Line by Line Solutions for Chapter 3: Determinants and Matrices provides a detailed, systematic explanation of each concept in the chapter. The chapter covers key topics such as the properties of determinants, the calculation of determinants, the concept of ad joint and inverse of matrices, and applications of matrices and determinants.

We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating the development of the theory of Toeplitz determinants.

Book Source: Digital Library of India Item 2015.4951 dc.contributor.author: Turnbull.h.w dc.date.accessioned: 2015-06-19T14:57:51Z dc.date.available: 2015-06-19T14:57:51Z dc.date.digitalpublicationdate: 2005-06-15 dc.date.citation: 1929 dc.identifier: Librarian, Bits-Pilani dc.identifier.barcode: 99999990149085 dc.identifier.origpath: /data1/upload/0004/182 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/4951 dc.description.scanningcentre: C-DAC, Noida dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 360 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Blackie And Son Limited,london dc.rights: Copyright Permitted dc.source.library: Central Library, Bits- Pilani dc.subject.classification: Generalities dc.subject.keywords: Transposed dc.subject.keywords: Permutation dc.subject.keywords: Convolutions dc.title: The Theory Of Determinants Matrices And Invariants

Book Source: Digital Library of India Item 2015.4951 dc.contributor.author: Turnbull.h.w dc.date.accessioned: 2015-06-19T14:57:51Z dc.date.available: 2015-06-19T14:57:51Z dc.date.digitalpublicationdate: 2005-06-15 dc.date.citation: 1929 dc.identifier: Librarian, Bits-Pilani dc.identifier.barcode: 99999990149085 dc.identifier.origpath: /data1/upload/0004/182 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/4951 dc.description.scanningcentre: C-DAC, Noida dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 360 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Blackie And Son Limited,london dc.rights: Copyright Permitted dc.source.library: Central Library, Bits- Pilani dc.subject.classification: Generalities dc.subject.keywords: Transposed dc.subject.keywords: Permutation dc.subject.keywords: Convolutions dc.title: The Theory Of Determinants Matrices And Invariants

Book Source: Digital Library of India Item 2015.4951 dc.contributor.author: Turnbull.h.w dc.date.accessioned: 2015-06-19T14:57:51Z dc.date.available: 2015-06-19T14:57:51Z dc.date.digitalpublicationdate: 2005-06-15 dc.date.citation: 1929 dc.identifier: Librarian, Bits-Pilani dc.identifier.barcode: 99999990149085 dc.identifier.origpath: /data1/upload/0004/182 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/4951 dc.description.scanningcentre: C-DAC, Noida dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 360 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Blackie And Son Limited,london dc.rights: Copyright Permitted dc.source.library: Central Library, Bits- Pilani dc.subject.classification: Generalities dc.subject.keywords: Transposed dc.subject.keywords: Permutation dc.subject.keywords: Convolutions dc.title: The Theory Of Determinants Matrices And Invariants

Let n\geq3 and J_{n}:=circ(J_{1},J_{2},...,J_{n}) and j_{n}:=\circ(j_{0},j_{1},...,j_{n-1}) be the n\timesn circulant matrices, associated with the nth Jacobsthal number J_{n} and the nth Jacobsthal-Lucas number j_{n}, respectively. The determinants of J_{n} and j_{n} are obtained in terms of the Jacobsthal and Jacobsthal-Lucas numbers. These imply that J_{n} and j_{n} are invertible. We also derive the inverses of J_{n} and j_{n}.

This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.

This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.

This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.

STEP FSC 1ST YEAR PART I XI MATHEMATICS LECTURES, DO NOT SELL!!, FREE FOR STUDIOUS STUDENTS. ALL COPYRIGHTS BELONGS TO STEP BY PGC

Lecture notes for the proceedings of the workshop "Algebraic Combinatorics related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S., Nara, Japan.

Lecture notes for the proceedings of the workshop "Algebraic Combinatorics related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S., Nara, Japan.

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.

In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of $d \times d$ matrices, if their traces are known. As a next step $4 \times 4$ matrices are expressed in terms of Dirac covariants. The third step is the calculation of the corresponding inverse matrices in terms of Dirac covariants.

In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of $d \times d$ matrices, if their traces are known. As a next step $4 \times 4$ matrices are expressed in terms of Dirac covariants. The third step is the calculation of the corresponding inverse matrices in terms of Dirac covariants.

In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.

This resource provides topic-wise, line-by-line questions for Chapter 3 of Class 12 Mathematics, Determinants and Matrices. It covers key concepts such as determinants, properties of determinants, evaluation of determinants, and applications of determinants (e.g., area of a triangle, solving linear equations). The chapter also focuses on matrices, including types of matrices, operations on matrices (addition, multiplication, transposition), and the inverse of a matrix. These questions are designed to help students understand matrix operations and determinant properties, essential for solving complex problems and excelling in exams.

This resource provides topic-wise, line-by-line questions for Chapter 3 of Class 12 Mathematics, Determinants and Matrices. It covers key concepts such as determinants, properties of determinants, evaluation of determinants, and applications of determinants (e.g., area of a triangle, solving linear equations). The chapter also focuses on matrices, including types of matrices, operations on matrices (addition, multiplication, transposition), and the inverse of a matrix. These questions are designed to help students understand matrix operations and determinant properties, essential for solving complex problems and excelling in exams.

The purpose of this article is to study the eigenvalues $u_1^{\, t}=e^{it\theta_1},\dots,u_N^{\,t}=e^{it\theta_N}$ of $U^t$ where $U$ is a large $N\times N$ random unitary matrix and $t>0$. In particular we are interested in the typical times $t$ for which all the eigenvalues are simultaneously close to $1$ in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first orders of the large $N$ asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge towards an exponential distribution when $N$ is large. Numeric simulations are provided along the paper to illustrate the results.

Let P=\circ(P_{1},P_{2},...,P_{n}) and Q=\circ(Q_{1},Q_{2},...,Q_{n}) be n\timesn circulant matrices where P_{n} and Q_{n} are nth Pell and Pell-Lucas numbers, respectively. The determinants of the matrices P and Q were expressed by the Pell and Pell-Lucas numbers. After, we prove that the matrices P and Q are the invertible for n\geq3 and then the inverses of the matrices P and Q are derived.

Source: Digital Library of India Scanning Centre: Allama Iqbal Library, University of Kashmir Source Library: Womens College,m .a Road Srinagar Date Accessioned: 9/18/2015 21:18 The Digital Library of India was a project under the auspices of the Government of India.

vii, 144 p. illus. ; 19 cm

The generalized sequence of numbers is defined by W_{n}=pW_{n-1}+qW_{n-2} with initial conditions W_{0}=a and W_{1}=b for a,b,p,q\inZ and n\geq2, respectively. Let W_{n}=circ(W_{1},W_{2},...,W_{n}). The aim of this paper is to establish some useful formulas for the determinants and inverses of W_{n} using the nice properties of the number sequences. Matrix decompositions are derived for W_{n} in order to obtain the results.

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener-Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the K\"ahlerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces.

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an arrangement of $N$ hyperplanes in an $n$-dimensional affine space. We assign weights $\al_1, ..., \al_N$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of $k$-dimensional hypergeometric integrals and the second is a matrix of $n$-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler's gamma function multiplied by a product of exponentials of the weights.

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an arrangement of $N$ hyperplanes in an $n$-dimensional affine space. We assign weights $\al_1, ..., \al_N$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of $k$-dimensional hypergeometric integrals and the second is a matrix of $n$-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler's gamma function multiplied by a product of exponentials of the weights.

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an arrangement of $N$ hyperplanes in an $n$-dimensional affine space. We assign weights $\al_1, ..., \al_N$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of $k$-dimensional hypergeometric integrals and the second is a matrix of $n$-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler's gamma function multiplied by a product of exponentials of the weights.

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an arrangement of $N$ hyperplanes in an $n$-dimensional affine space. We assign weights $\al_1, ..., \al_N$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of $k$-dimensional hypergeometric integrals and the second is a matrix of $n$-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler's gamma function multiplied by a product of exponentials of the weights.

Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8.

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Let $\mathcal{V}=\bigsqcup_{i=0}^n\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and let $\mathcal{A}$ be the graded Gorenstein algebra defined over $\mathbb{Q}$ which has $\mathcal{V}$ as a $\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\mathcal{A}$. We compute explicitly the Hessian determinant $|\frac{\partial ^2F}{\partial X_i \partial X_j}|$ evaluated at the point $X_1 = X_2 = \cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\mathcal{V}_1$ and $\mathcal{V}_{n-1}$. Our exploration is motivated by the fact that both of these matrices arise naturally in the study of the Sperner property of the lattice and the Lefschetz property for the graded Artinian Gorenstein algebra associated to it.

The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.

In this paper we study a collections of operators $a(k)$ satisfying the "$q_{kl} $-canonical commutation relations" $a(k)a^{+}(l)-q_{kl}a^{+}(l)a(k) =\delta_{kl} $ (corresponding for $q_{kl}=q$ to Greenberg (infinite) statistics, for $q=\pm 1$ to classical Bose and Fermi statistics).We show that $n!\times n!$ matrices $A_{n}(\{q_{kl}\})$ of scalar products of n-particle states is positive definite for all n if $|q_{kl}|

We prove a determinantal identity concerning Schur functions for 2-staircase diagrams lambda=(ln+l',ln,l(n-1)+l',l(n-1),...,l+l',l,l',0). When l=1 and l'=0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined enumerations of Alternating Sign Matrices.

We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determinant, modulo a factor n!

We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determinant, modulo a factor n!

x, 225 p. 24 cm

Source: Digital Library of India Scanning Centre: IISc, Bangalore Source Library: Jrd Tata Library,indian Institute Of Science,bangalore Date Accessioned: 7/9/2015 19:50 The Digital Library of India was a project under the auspices of the Government of India.

An expansion of the determinant of any matrix in terms of row sums and off-diagonal entries is given and used to obtain expressions for the coefficients of the characteristic polynomial of stochastic matrices.

We study the limiting behavior of $\Tr U^{k(n)}$, where $U$ is a $n\times n$ random unitary matrix and $k(n)$ is a natural number that may vary with $n$ in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szeg\"o limit theorem for Toeplitz determinants associated to symbols depending on $n$ in a particular way. As a consequence to this result, we find that for each fixed $m\in \N$, the random variables $ \Tr U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $j=1,..., m$, converge to independent standard complex normals.

We show that the apolar ideals to the determinant and permanent of a generic matrix, the Pfaffian of a generic skew symmetric matrix and the Hafnian of a generic symmetric matrix are each generated in degree two. In each case we specify the generators and a Gr\"{o}bner basis of the apolar ideal. As a consequence, using a result of K. Ranestad and F. O. Schreyer we give lower bounds to the cactus rank and rank of each of these invariants. We compare these bounds with those obtained by J. Landsberg and Z. Teitler.

Book Source: Digital Library of India Item 2015.205508 dc.contributor.author: Aitken. A.c dc.date.accessioned: 2015-07-09T14:09:42Z dc.date.available: 2015-07-09T14:09:42Z dc.date.digitalpublicationdate: 2005-01-11 dc.date.citation: 1944 dc.identifier.barcode: 1040010074700 dc.identifier.origpath: /data_copy/upload/0074/705 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/205508 dc.description.scannerno: 1 dc.description.scanningcentre: IISc, Bangalore dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 143 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Svinfo dc.publisher: Oliver And Boyd,london dc.source.library: Jrd Tata Memorial Library, Indian Institute Of Science, Bangalore dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Number Theory dc.subject.keywords: Fundamental Operations Of Matrices dc.subject.keywords: Properties Of Determinants dc.title: Determinants And Matrices

We show that the apolar ideal to the determinant of a generic symmetric matrix is generated in degree two, and the apolar ideal to the permanent of a generic symmetric matrix is generated in degrees two and three. In each case we specify the generators of the apolar ideal. As a consequence, using a result of K. Ranestad and F. O. Schreyer we give lower bounds to the cactus rank and rank of each of these polynomials. We compare these bounds with those obtained by J. Landsberg and Z. Teitler.

Matrices and Determinants gb sir

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