Questions tagged [determinant]

Questions about determinants: their computation or their theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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Determinant of matrix with binomial coefficient elements equal to $\binom{2n-1}{n}$

Given an $n \times n$ matrix $A_n$ with elements $a_{ij}=\binom{n+i-1}{j}$, $1 \le i \le n$, $1 \le j \le n$, I noticed that its determinant $\lvert A_n \rvert$ seem to satisfy: $$\lvert A_n \rvert = \...
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Given matrix $A \in \Bbb R^{n \times n}$ such that $A^2=-I$, find $\det(A)$

Given an $n \times n$ matrix $A$ with real entries such that $A^2=-I$, find the $\det (A)$. My Attempt: $|A|^2=(-1)^n\implies|A|=(-1)^{\frac n2}$ The answer given is $|A|=1$ Eigenvalues are not ...
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Determinant of triangular matrix with extra diagonal

If we have a triangular matrix we can calculate the determinant in $O(n)$. If we have a triangular matrix with one extra diagonal above the main diagonal, so for example: \begin{Vmatrix} a_1 & a_2 ...
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Prove by determinant properties

[(ax+by,ay+bx,az+bx)(ay+bz,az+bx,ax+by)(az+bx,ax+by,ay+bz)]=(a^3+b^3)[(x,y,z)(y,z,x)(z,x,y)]​ have been trying to solve this but l don't know what to do with a+b.would like to know how to solve this ...
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Showing that if the parametrizing values $a_1,\dots,a_n$ of a Vandermonde matrix are unique then it has a non-zero determinant with the use of the FTA

My whole title would be: Showing that if the parametrizing values $a_1,\dots,a_n$ of a Vandermonde matrix are unique then it has a non-zero determinant, with the use of the fundamental theorem of ...
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Similar matrices and their dets [closed]

suppose that the nxn matrices A and B are similar and that each has n real eigen values. Show that detA = detB
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Prove that a linear partial differential equation of first order can't be elliptic

A linear PDE of first order has the general form $$a(x,y)u_x+b(x,y)u_y-c(x,y)u-d(x,y)=0$$ A PDE is elliptic if the symmetric matrix of the coefficients of the highest derivatives has a determinant ...
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1 answer
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If a skew-symmetric real matrix has all eigenvalues zero, must it be the zero matrix?

This can be easily verified for $2\times2$ and $3\times3$ matrices, but can the result be generalised?
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1 answer
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The characteristic polynomial of this family of matrices

I'm looking at the following family of $n\times n$ matrices. The entries are 0 everywhere except above and below the diagonal. Above it takes values from $1 \to n-1$ and below from $ -n +1 \to -1$. ...
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3 votes
1 answer
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If $A$ is a symmetric matrix, then $\det(A) \leq \prod\limits_{i = 1}^d a_{ii}.$

Prove or provide a counterexample. If $A = (a_{ij})$ is a symmetric matrix, then $\det(A) \leq \prod\limits_{i = 1}^d a_{ii}.$ The result is obviously true for diagonal matrices and here is a proof ...
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find a formula for the square of the product of two determinants

I was wondering how one could come up with the matrices described in the solution below, if say one were to solve the above problem for the first time?
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3 votes
3 answers
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Is there a method to calculate the eigenvalues for this a n x n symmetrical matrix

I'm working with a mechanics problem where I try to find the eigenmodes of the system. The system contains of $n$ masses all connected with springs to one another (same spring constant $k$), the outer ...
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Calculate: $\Delta=\left|\begin{array}{ccc} b c & c a & a b \\ a(b+c) & b(c+a) & c(a+b) \\ a^{2} & b^{2} & c^{2} \end{array}\right|$

Calculate: $$\Delta=\left|\begin{array}{ccc} b c & c a & a b \\ a(b+c) & b(c+a) & c(a+b) \\ a^{2} & b^{2} & c^{2} \end{array}\right|$$ Does anyone know any easy way to ...
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Is the relation $\det M_n = \prod_i \lambda_i$ general? [duplicate]

I read in my book that the eigenvalues of a matrix $M_n$ which size is $n$ are linked by: $$\det M_n = \prod_i^n \lambda_i$$ This is quite usufull when $n = 2$ and $\det M = 1$. I wondered if this ...
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How can I expand this expression to arrive at this determinant? [duplicate]

I need to expand this expression $$\sum_{j=1}^{n}\prod_{\substack{j\neq k>l\neq j\\k,l=1}}^n(\lambda_k-\lambda_l)(-1)^j$$ to get this $$\begin{equation} \begin{vmatrix} 1 & 1 & \lambda_1 &...
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Area of a crossed diagonal quadrilateral

If four coordinates of vertices are given, the area of the first convex quadrilateral is expressed in known standard matrix form. How is the net (positive and negative sum ) area expressed for the ...
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1 answer
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Linear combination and sign of determinant

Given $d+2$ vectors $x_1,x_2,\dots,x_{d+2} \in \mathbb{R}^d$. Clearly, they are linearly dependent. However, we have better result: Let $$ y_i = \begin{bmatrix} x_i\\ 1 \end{bmatrix} \in \mathbb{R}^{d+...
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1 vote
1 answer
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Upper bound on trace for powers of a matrix which converges to zero

For some fixed square matrix $X$ for which $\lim_{n\rightarrow\infty} X^n = 0$, it is apparent that the trace of $X^n$ must also converge to zero as $n\rightarrow\infty$. Is there anything I can do to ...
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3 votes
2 answers
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How can I prove that $f(x)=\det (A+xB)= \alpha x+\beta $?

I have two matrices $A$ and $B$, such that : $$A=A(a,b,c)=\begin{pmatrix} a & c & c & \dots & c \\ b & a & c & \dots & c\\ b & b & a & \dots & c\\ \...
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1 vote
1 answer
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Determinant of circulant $(0,1)$ matrices of certain form

I am interested in computing the determinant of the following circulant matrices: let $n=p^k$ for $p$ a prime and $k\in \mathbb{N}$, take $a\in \mathbb{N}$ to be such that $a<p$ and $(a,p)=1$. ...
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Does the Nullity Theorem hold in fields of characteristic 2?

I'm playing around with involutory ($M^2 = I$) matrices over finite fields with characteristic 2 ($\mathbb{F}_{2^m}$). I came across the nullity theorem, which seems very useful to check if ...
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1 answer
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Rewrite proposition with logical symbols

I want to rewrite the following proposition in mathematical language (and by mathematical language I mean symbols such as: $\forall , \exists, (, ), \implies$ and so on). Proposition: Every non-...
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2 votes
2 answers
50 views

Determinant of the linear transformation $S: \mathbb{K}^{n\times n} \to \mathbb{K}^{n\times n}$ such that $S(X) \mapsto X^t$

$\mathbb{K}$ is a field and $n \geq 1$. Let $S: \mathbb{K}^{n\times n} \to \mathbb{K}^{n\times n}$ such that $S(X) \mapsto X^t$ be a linear transformation. What is the determinant of $S$? I know that $...
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2 answers
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If $A$ is a square matrix

If $A$ is a square matrix, and $A=A^2$, then what would the possible values of $|A|$? I've tried to calculate it through basic mathematics, however I feel it's not appropriate... $$A=A^2$$ $$A-A^2=0$$ ...
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Homogenous system of equations non trivial solution condition checking

This was written in a text : Theorem: The homogeneous linear system $$ \begin{array}{l} a_{1} x+b_{1} y+c_{1} z=0 \\ a_{2} x+b_{2} y+c_{2} z=0 \\ a_{3} x+b_{3} y+c_{3} z=0 \end{array} $$ has a non-...
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Help calculating this determinant using induction

I want to show that the determinant of the $(n \times n)$ matrix $$J(\alpha)=\begin{pmatrix} -\lambda & \frac{1}{2\alpha} & \\ \frac{\alpha}{2} & -\lambda & \\ &&\ddots&\...
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1 answer
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What concise function $f$ satisfies $\prod_{i=1}^n \prod_{j=1}^n a_{i,j} = f \left( \det(\mathbf{A}) \operatorname{Adj}(\mathbf{A})^{-1} \right)$?

Let $\mathbf{A} = [a_{i,j}]$ be an $n \times n$ matrix with determinant $\det (\mathbf{A})$. Consider the equality $$\prod_{i=1}^n \prod_{j=1}^n a_{i,j} = f \left(\det(\mathbf{A}) \operatorname{Adj}(\...
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Is $|\bf A \bf B^{-1} + \bf I| \geq |\bf A (\bf B + \bf C)^{-1} + \bf I|$ true?

$\bf A$, $\bf B$, and $\bf C$ are Hermitian matrices of the same order. They satisfy $\bf A \succeq \bf 0$, $\bf B \succ \bf 0$, and $\bf C \succeq \bf 0$. Then, is \begin{equation} |{\bf A} {\bf B}^{-...
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2 answers
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Jacobian identity used in proof of change of variables

This is from "Calculus on Manifolds", proof of Change of Variables theorem. I don't understand why these two red circled expressions are equal. $$|det(h\circ g)'|=|det(h'\circ g)||det(g')|$$ ...
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Given the value of a determinant what is the quickest way to calculate the value of a determinant which has only one column different?

I have a determinant $ \begin{vmatrix} a_{11}\ a_{12}\ a_{13} .... a_{1n}\\ a_{21}\ a_{22}\ a_{23} .... a_{2n}\\ .................\\ .................\\ a_{n1}\ a_{n2}\ a_{n3} .... a_{nn}\\ \end{...
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0 answers
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Is this an incorrect application of Tutte's theorem of perfect matching for bipartite graphs?

This is an extract from a conference paper. It seems the authors are invoking Tutte's theorem (since [12] refers to the 1947 paper) to conclude that a matrix $J(x)$ with given numerical entries is ...
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1 answer
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Tutte's matrix for perfect matching in bipartite graphs

I came across Tutte's matrix for a bipartite graph $G(U, V, E)$ in two different forms. One form (seen in these notes for example https://www.cs.cmu.edu/afs/cs/academic/class/15859-f04/www/scribes/...
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1 answer
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Determinant of baricenter

I want to know whether it is easy to calculate the below $\det$ or not. $$\operatorname{det}((1-\alpha) M+\alpha \mathrm{Id}),$$ where $M=\left[\begin{array}{cccc} *_{n_{1}} & * & \cdots & ...
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Bound for volume difference between translated parallelotopes formed between 2 halfspaces

Consider 2 d-dimensional halfspaces given by $h_1$ : $\langle w,x\rangle < a$ $h_1$ : $\langle w,x\rangle < b$ where $w,x \in \mathbb{R}^d$ and $|a-b| \leq \Delta$. Consider a hypercube $C \in [...
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Determinant ratio approaching one-half [duplicate]

Let $\det M(n)$ be the generalization of this $5 \times 5$ determinant: $$ \begin{vmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \\ 3 & 4 & 5 & 1 & ...
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1 vote
1 answer
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Determining a matrix's invertibility with eigenvalues

Let $A$ be a 4 x 4 matrix with eigenvalues -4, -2, 4, -3. Then, the determinant of $A$ is the product of all the eigenvalues, which is, -96. So, how does one determine if a matrix $A-4I$ is invertible ...
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2 votes
1 answer
82 views

If $\det[v_1 \mid v_2 \mid v_3] = 2$, what is the value of $\det[v_{1} + v_{2} + v_{3} \mid v_{1} + 2v_{2} + 3v_{3} \mid v_{1} + 4v_{2} + 9v_{3}]$?

Let $A$ = $\begin{bmatrix} v_1 \hspace{2 mm} v_2 \hspace{2 mm} v_3 \end{bmatrix}$ be a 3 x 3 matrix with column vectors $v_1, v_2, v_3$. If $det(A) = 2$, then what is $det \begin{bmatrix} v_1+v_2+v_3 \...
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0 answers
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Coefficients of characteristic polynomial

I am studying Halmos' Finite-dimensional vector spaces, Section 82. The following is what I understand. We can express the property of positiveness of a matrix $(\alpha_{ij})$ as $$\langle A x ,x \...
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1 answer
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Determinant of a special diagonal matrix

I have a matrix of: $$ A = \begin{bmatrix} a & b & b & \cdots & b \cr -1 & 1 & 0 & \ddots & \vdots \cr 0 & -1 & 1 & \ddots & \vdots \cr \vdots & \...
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0 answers
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Invertibility of the Gram matrix of convex combination

I am struggling with this this question: Let assume two real valued matrices $A,B\in R^{w\times d}$, which $w>d$ and they both have full (column) rank. Now, I am interested to study invertibility ...
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3 votes
1 answer
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Existence of an integer matrix with maximal subdeterminants $a_1, \ldots, a_n$

Given $n \geq 2$ and integers $a_1, \ldots, a_n$, does there exist an integer $(n-1) \times n$ matrix whose maximal subdeterminants are $a_1, \ldots, a_n$ (with fixed ordering)? Example: $n = 3$, $(...
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Volume of a particular $n$-dimensional region

What is the $n$-dimensional volume of the region $\{ x \in \mathbb{R}^n|x_i \geq 0$ for all $i = 1,...,n$ and $x_1 + ... + x_n ≤ 1 \}$ I'm thinking of using a recurrence relation but I don't know how ...
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0 votes
1 answer
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Determine if LU decomposition is possible on a matrix?

I am trying to understand how you determine if LU decomposition is possible on a given matrix. I believe the way to calculate this is to check if the leading-matrices have non-zero determinants. I ...
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Hankel determinants related to Motzkin paths

Let $M_{n,k}(t,s)$ denote the weight of all Motzkin paths from $(0,0)$ to $(n,k)$ where the weight of a horizontal step at height 0 is $s,$ the weight of a horizontal step at height $k\geq 1$ is $t$ ...
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Is my proof about the rank of a matrix correct?

The problem: Let $A\in\mathbb{K}^{m\times n}$ be a matrix. Show that $\text{rank}({A})$ is identical to the largest number $k\in\{1,2\dots,\min(m,n)\}$ such that the subdeterminant with $i_1<i_2<...
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Determining value of constants in simultaneous equations using inverse matrix when determinant is zero

I've been having trouble understanding how to solve this problem: Determine the values of the real constants a and b for which there are infinitely many solutions to the simultaneous equations 2x + 3y ...
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Determinant inequality with unitary matrix

I come up with the following conjecture while doing my research, which is a determinant inequality. I have tried to run the MatLab simulation to verify its sanity. It seems that the inequality is true....
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Calculate determinant of expression with given determinant

I need to calculate the following determinant. Given $det(B) = \frac{1}{3}$ Calculate: $det(AB^{T}B^{-1}A^{T}B^{4}(A^{-1})^{T}(A^{T})^{-1})$ My solution: $det(A \cdot \frac{1}{3} \cdot 3 \cdot A^{T} \...
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How many $3\times3$ integer determinants are a multiple of $10^p$?

I have $A$, a $3\times3$ matrix, the elements $a_{ij}\in\mathbb{N}$ and $0\le a_{ij}\le W$, $W\in\mathbb{N}$ (for me $0\in\mathbb{N}$). What is the number $N$ of $3\times3$-matrices with the bounds ...
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3 votes
1 answer
30 views

Intuitions behind Frobenius' generalization of characters to nonabelian finite group given the historical context

I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius. According to most of the related papers (e.g. Pioneers of ...
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