Question about determinants, computation or 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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38 views

Alternate formula for the determinant of a 3x3 Matrix

So, I've recently been reading about Dodgson condensation, for matrices. Take, in particular the $3x3$ case. Define a matrix, $|A|:$ \begin{pmatrix} a & b & c \\ d &...
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1answer
83 views

What is an intuition behind permanent?

I would like to know what is your intuition behind permanent of a matrix. For me, it looks like someone came and saw determinant, deleted permutation sign and voila, we have permanent and it counts ...
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2answers
70 views

Determinant of $N \times\ N$ matrix

So the question asks: For $n \geq 2$, compute the determinant of the following matrix: $$ B = \begin{bmatrix} -X & 1 & 0 & \cdots & 0 & 0 \\ ...
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1answer
39 views

Prove the theorem of row swapping determinants??

I don't know how to prove this theorem in a clear way, I could really use some help, Thanks so much! This is a linear-algebra problem dealing with determinants. Let M' be the matrix obtained from ...
2
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2answers
49 views

In which situations $\det(A\mod x) \mod x=\det(A)\mod x$ would help us knowing if $\det(A)=0$?

André Nicolas, in his very neat answer to is the following matrix invertible? uses the fact that the matrix $$ \begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 ...
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33 views

Finding Factors of a Determinant

Consider the determinant with elements: $a_{11} = ax-by-cz, a_{12}=ay+cz, a_{13}=cx+az$ $a_{21}=ay+bx, a_{22}=by-cz-ax, a_{23}=bz+cy$ $a_{31}=cx+az, a_{32}=bz+cy, a_{33}=cz-ax-by$ Where $a_{ij}$ ...
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1answer
28 views

Jacobian of a diffeomorphism

Let $U,V\subseteq \mathbb{R}^{n}$ be open. Let $\alpha:U \to V$ be a smooth homeomorphism. Furthermore, assume that $\mathcal{J}_{\alpha}(\mathbf{x})$ (the Jacobian matrix) has rank $n$ for all $\...
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48 views

A brief answer for the determinant of a matrix

I am given the following matrix $A=(a_{ij})_{6 \times 6}$, where $a_{ij}=\sum_{k=1}^{10} x_k^{i+j-2}$. Remark: If $A=(a_{ij})_{10 \times 10}$ with the same $a_{ij}$ defined above, the answer is very ...
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1answer
64 views

What is the average determinant of a matrix in $M_{2}(\mathbb{Z}/n\mathbb{Z})$?

Given a finite collection of matrices, it is natural to consider the average determinant of a matrix in such a collection. For example, an interesting problem involving the average determinant of a ...
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33 views

Matrices and cofactors question

From Mathematical Methods for Engineers and Scientists 1 by K.T. Tang: Example 4.6.1. Evaluate $$D_4=\begin{vmatrix}2&1&3&1\\1&0&2&5\\2&1&1&3\\1&3&0&...
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2answers
81 views

For a matrix $A$, does $A^3=I_n$ mean that $\det(A)=1$?

Here, $A\in \mathbb{K}^{n,n}$ ($n$ by $n$ matrix) where $\frac{1}{2} \in \mathbb{K}$ where $\mathbb{K}$ is a field. I think this is true. Since, using the fact that $\det(AB)=\det(A) \cdot \det(B)$ ...
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1answer
130 views

Tricky problem, looking at linear algebra

I have encountered a question to which I have found an algebraic answer yet am still gasping for a geometric one (I want to know why the answer is so beautiful!) Here it is: Find $\det(δ_{ij} +...
2
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2answers
43 views

Determinat of block matrix related

How to find determinant of following block matrix? $\begin{bmatrix} A & A \\ A & kI \end{bmatrix}$ Where $A$ is any square matrix,$I$ is an identity matrix and $k$ is any constant.
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1answer
52 views

Doubt in the proof of Cauchy-Binet formula.

I have a doubt in the proof of cauchy binet formula. Question: In the colored portion. I dont know how the sum multiplying $det(A(J'))$ become $$\sum_\sigma sgn(\sigma) b_{j'_{\tau(1)}1} b_{j'_{\...
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1answer
99 views

Evaluation of the determinant of a special $n\times n$ matrix

Let us consider rational function $A(x,n)$ ($n\in\mathbb{N}$, $x\in D=\mathbb{R}\setminus \{-1,1\}$): $$A(x,n)=\det\begin{pmatrix} 1 & [1]&0&0& \ldots & 0 &0\\ x^{1\times2} &...
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1answer
18 views

Showing that a matrix is singular

Consider a matrix $A$ of dimension $4\times 4$ $$ A=\begin{pmatrix} a_{1} & a_2 & a_3 & a_4\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 &...
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31 views

Proof of result involving Pfaffian of a matrix

Show that $$\text{Pf} MAM^T = \text{det}M \cdot \text{Pf} A$$ for any matrix $M$ and antisymmetric $A$. Attempt: $$\text{Pf} MAM^T = \frac{1}{2^N N!} \epsilon_{\alpha_1 \dots \alpha_{2N}} (MAM^T)_{\...
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18 views

Question conerning the determimant of the sum of two matrices- revised format

Sorry for the precedent "version" of my question. Here (in an attached file) there is a complete version of this question I have concerning the determinant of the sum of two matrices. I'm sorry ...
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19 views

Prove if the determinant of a matrix is positive, then it has a Cholesky factorization

Prove if the determinant of a matrix is positive, then it has a Cholesky factorization. A cholesky factorization can only be performed for hermitian, positive definite matrices. Should I go about ...
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26 views

determinant equality [closed]

Prove that if $ A,B,C\in M_{2}(\mathbb{R}) $ so that $ A^{2}+B^{2}+C^{2}=AB+BC+CA, $ then $ det(AB-BA+BC-CB+CA-AC)=0. $
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2answers
63 views

Transformation of positive semi-definite matrices

Let $a,b,c,d,e$ be positive reals such that the following matrix is positive semi-definite: $$ \begin{pmatrix} a+4b+6c+4d+e & a+3b+3c+d & a+2b+c \\ a+3b+3c+d & a+2b+c & a+b \\ a+2b+...
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186 views

How to tackle this polynomial given as a determinant? [closed]

Let $$p (x) = \begin{vmatrix} 1 & x & x & \dots & x & x \\ x & 1 & x & \dots & x & x \\ x & x & 1 & \dots & x & x \\ \vdots & \vdots &...
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1answer
48 views

Prove that $\det V(1,2,..,n)$ divides $\det V(k_1,k_2,…,k_n)$, where each $V$ is the associated Vandermonde matrix

Let $k_1<k_2<...<k_n$ be integer numbers. Prove that $$\det\big(V_n(1,2,..,n)\big)\mid \det\big(V_n(k_1,k_2,...,k_n)\big),$$ where V is a Vandermonde matrix. I think it is a generalization ...
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108 views

Determinant of $\vert A' C A\vert$

Let $X$ be an $n\times p$ real matrix with column rank $k$, where $0<k<p<n$, and let $A$ be a semi-orthogonal matrix (the columns are orthonormal) such that $A'X=0$, i.e. the column space of $...
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1answer
82 views

Describe the set of all points x1, x2 such that the determinant is zero.

Consider three distinct points $(a_{1}, a_{2}), (b_{1}, b_{2}), (c_{1}, c_{2})$ in the plane. Describe the set of all points $(x_{1}, x_{2})$ satisfying the equation: $$det\begin{pmatrix}1&1&...
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1answer
45 views

Determinants and Trigonometry

If $$ \det \begin{bmatrix} \sin(2x) & \cos^2 x & \cos(4x) \\ \cos^2 x & \cos(2x) & \sin^2 x \\ \cos^4 x & \sin^2 x & \sin(2x) \end{bmatrix} = A + B\sin x + C \sin^2 x + \cdots +...
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1answer
56 views

Determinant of Matrix $n \times n$ defined using cosine function

In class, the professor asked us to find the determinant of the matrix $$\left(\cos(i+j-1)\right)_{1\leq i,j\leq n}=\begin{pmatrix}\cos(1) & \cos(2) & \cdots & \cos(n)\\ \cos(2) & \cos(...
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38 views

Prove that for each real number $r$ we can find an $n \times n$ square matrix $A$ with real entries such that determinant of $A$ is $r$.

I have proved that for each positive real number we can find an $n \times n$ diagonal matrix with each diagonal entries $r^{1/n}$ such $\det A=r.$ But how to prove for negative real numbers?
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1answer
28 views

Gaussian integration involving operators

I was just wondering if someone could explain the following series of equalities: The Gaussian integral may be evaluated using an orthogonal transformation $R$ to diagonalise the real symmetric ...
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32 views

Find the inverted matrix of $A=(a_i,_j),a_{i,j}={\dbinom{j-1}{i-1}}$

Let be $A=(a_i,_j)\in M_{n+1}(\mathbb{R})$ defined for all $(i,j)\in [\![ 1,n+1 ]\!]^2$, by $a_{i,j}={\dbinom{j-1}{i-1}}$. Let's show this is invertible and determine its inverted matrix. To my mind,...
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2answers
31 views

Determinant of a portioned matrix

If $A$ and $B$ are $n\times n$ matrices and $C$ is defined to be $$ C=\begin{pmatrix} O&A\\ B&O \end{pmatrix} $$ Where $O$ denotes the zero matrix. Can I conclude that $O$ needs to be only ...
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1answer
59 views

How to prove or disprove matrix $A$ is invertible $\iff$ $\det{A}\ne 0$ if it's defined over field $F_p$

How to prove or disprove matrix $A$ is invertible $\iff$ $\det{A}\ne 0$ if it's defined over field $F_p$ What's special about $A$ if it's defined over field $F_p$? For normal procedure, we have $A$ ...
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86 views

Assign a unique number to every linear form $\varphi : \mathbb R^n \to \mathbb R$, has this number a geometric interpretation?

If $V$ is a finite-dimensional vector space of dimension $n$, denote the space of all alternating $k$-fold multilinear maps (also called alternating $k$-tensors) by $\Lambda^k(V)$. Then $\dim \Lambda^...
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1answer
81 views

Determinant of a symmetric, sparse matrix

I'm trying to calculate the determinant of the $q \times q$ matrix shown here: $$L(y)=\begin{bmatrix} V(1) & 1 & 0 & \cdots & 0 & 1 \\ 1 & V(2) & 1 & \ddots & \...
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1answer
36 views

Show that similar matrices have the same characteristic polynomial

I am given that $A$ is a square matrix and $B=C^{-1}AC.$ I'm trying to show that $A$ and $B$ have the same characteristic polynomial. So far I have said the following: $$p_A(\lambda)=\det(A-\lambda ...
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2answers
70 views

On the definition of the volume form in general vector spaces as given in Spivak, Calculus on Manifolds

For a vector space $V$ denote by $\Lambda^k(V)$ the space of alternating $k$-tensors, or alternating $k$-fold multilinear maps on $V$. I have some difficulty following the intention of the author in ...
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23 views

Application of the Leibniz formula

I am trying to show that in the characteristic polynomial of $\det (A-\lambda I)$ the coefficient of $\lambda^{n-1}$ is $(-1)^{n-1}TrA.$ I'm given the hint to use the Leibniz formula for this ...
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2answers
56 views

How does the determinant link to the cross product

For a $2\times 2$ matrix $$\begin{pmatrix}a&b\\c&d \end{pmatrix} $$ The determinant is given by $ad-bc$. And the cross product of $$\begin{pmatrix} a\\b\\0\end{pmatrix}\times \begin{pmatrix} c\...
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1answer
333 views

What is the number of $n \times n$ binary matrices $A$ such that $\det(A) = \text{perm}(A)$?

Recall that the permanent is the 'positive analog' of the determinant whereby the signs in the cofactor expansion process are taken as positive. That is, the permanent is the immanant corresponding to ...
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3answers
100 views

Determinant of $2 \times 2$ matrix such that $A = A^{-1}$

Let $A$ be a $2 \times 2$ matrix such that $A = A^{-1}$. The value of $\operatorname{det} (A)$ can be: $\operatorname{det} (A)=-2$ $\operatorname{det} (A)=-1$ $\operatorname{det} (A)=0$ $...
3
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105 views

$2\times 2$ block Toeplitz determinant

My question is about computing asymptotic the determinant (dimension of the matrix $n\to\infty$) of a $2\times 2$ block Toeplitz matrix. $$\mbox{det}\left(\begin{array}{cc} a_n & b_n \\ d_n & ...
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1answer
48 views

How to derive the Vandermonde Determinant?

I watched this video https://www.youtube.com/watch?v=87iJTcXqTKY explaning the Vandermonde Determinant I understood everything but I was wondering why the guy never mentioned the (-1)^(i+j) term used ...
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1answer
55 views

Proof about determinant for a special matrix?

Given a $n\times n$ matrix $$A= \begin{pmatrix} a_{1}+p&p&p&p&\cdots & p&p&p\\ p&a_{2}+p&p&p&\cdots&p&p&p\\ \vdots & \vdots & \...
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35 views

pivot formula for determinant

This is my textbook what's the name of this formula? I don't understand ifthe diagonal entries are pivots which require the leading entry to be 1, but matrix U is just echelon form rather than ...
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40 views

Find determinant of cofactor matrix

If have a matrix $A$ that all I know about is its size and what its determinant is? For example a $4\times4$ matrix with a determinant of $3$. How can I find the determinant of the cofactor matrix $...
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50 views

The derivative of a determinant

Let $f(A)=detA$ for all A in $L(\mathbb{R}^n\rightarrow \mathbb{R}^n)$ Prove: a) $f(A)=Det(A)$ is a continuously differentiable function. b)$(Df)_{Id}(H)=tr(H)$ for all $H\in L(\mathbb{R}...
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2answers
90 views

Determinant $\left|\begin{smallmatrix} y+z &x &y \\ z+x &z &x \\ x+y &y &z \\ \end{smallmatrix}\right|= (x+y+z)(z-x)^2$

Tomorrow I have to attend a math exam. So I have to prove a problem on determinant. The Problem: $$\left|\begin{matrix} y+z &x &y \cr z+x &z &x \cr x+y &y &z \cr \end{matrix}\...
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2answers
81 views

How to prove that a square matrix's determinant is zero given its row and column properties? [duplicate]

How can I prove the following? If $A$ is an $n \times n$ matrix such that $$ \sum\limits_{j=1}^n a_{ij} = 0 $$ for $1 \leq i \leq n$ then $\det A = 0$.
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5answers
120 views

Prove that the determinant is a multiple of $17$ without developing it

Let, matrix is given as : $$D=\begin{bmatrix} 1 & 1 & 9 \\ 1 & 8 & 7 \\ 1 & 5 & 3\end{bmatrix}$$ Prove that the determinant is a multiple of $17$ without developing it? ...
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1answer
29 views

A different determinant expansion

Let us denote the determinant of a matrix $A=((a_{ij}))_{i,j=1}^n$ as $\displaystyle \det_{i,j=1}^n a_{ij}$. Let $\delta_{ij}$ be the kronecker delta function and suppose $a_{ij}=a_{ji}$ for all ...