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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1answer
52 views

Expected number of times to get arbitrary arrangement of coins

I'm thinking about a question: We consider tossing coins repeatedly. Using $+1$ to denote front and $-1$ back, given a positive interger $m$ and $\sigma=(\sigma_1,\dots,\sigma_m)$ where $\sigma_i\in\...
9
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5answers
155 views

Compute $\det{T}$ where $T(X)=AX+XA$

Consider the linear transformation $T:V\to V$ given by $T(X) = AX + XA$, where $$A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}.$$ Compute the determinant $\det T$. ...
1
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1answer
26 views

Determinant comparison about skew-symmetric matrices

Suppose $S$ is a real skew-symmetric matrix, show that $\det(I+S) \geq 1$, where equality holds iff $S=0$. My idea is to define a function $f(t)=\det(I+tS)$, for a fixed $S \neq 0$, and then show ...
0
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1answer
40 views

Determinant of $e^A$. [duplicate]

Suppose $A$ is a matrix in $\mathbb{R}^{k \times k}$. Show that $$\det e^A = e^{\text{Tr } A},$$ where $\text{Tr } A$ is the trace of the matrix $A$. This is marked as a "starred exercise" in ...
0
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1answer
54 views

Properties of determinants

Prove using properties of determinants : \begin{equation*} \left|\begin{matrix} b^2 + c^2 & a^2 & a^2\\ b^2 & c^2 + a^2 & b^2\\ c^2 & c^2 & a^2 + b^2 \end{matrix}\right| = 4a^...
3
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2answers
67 views

Prove two complex matrices have null trace

Let $A,B \in \mathbb{C}^{2 \times 2} \setminus \{O_2\}$, where $AB=-BA$ and $\det(A+B)=0$. Prove that $\operatorname{tr}(A) = \operatorname{tr}(B) = 0$ (where $\operatorname{tr}$ is the trace). My ...
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1answer
26 views

Constructing a determinantal inequality

The following is from page 3410 of the paper Quadratically constrained attitude control via semidefinite programming. Consider a polynomial: $$\mu_1(p_1^Tx)^2+ \cdots + \mu_n(p_n^Tx)^2\leq a$$ ...
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0answers
9 views

How does this alternative formulation of an hyperplane work?

I am studying 0/1 polytopes from Ziegler's lectures on polytopes https://arxiv.org/pdf/math/9909177.pdf. I found a small part of a proof of a corollary, which I do not understand. Here it is (...
2
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3answers
73 views

Row replacement operation not changing the determinant

Can someone prove why a row replacement operation does not change the determinant of a matrix? **row replacement operation being adding one row to another or something of that sort
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1answer
79 views

Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then ...
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1answer
45 views

Determinant of a large block matrix

$\newcommand{\lmt}{\left[\begin{matrix}}$ $\newcommand{\rmt}{\end{matrix}\right]}$ Hi, I was reading through a proof of the number of domino tilings of a $(2n)\times(2n)$ chessboard, and somewhere ...
0
votes
1answer
31 views

Prove that $\text{det}(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$

Let $f(x)=(p_1-x)\cdots (p_n-x)$ $p_1,...p_n\in \mathbb R$ and let $a,b\in \mathbb R$ such that $a\neq b$ Prove that $\text{det} A={bf(a)-af(b)\over b-a}$ where $A$ is the matrix: $$\begin{pmatrix}...
9
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1answer
101 views

Prove the n-th power of a matrix is the null matrix

Let $A,B$ squared matrixes with complex elements, $dim(A)=dim(B)=n, AB=BA, \det(B)\ne0$, having the following property: $|\det(A+zB)|=1, \forall z \in \mathbb{C}, |z|=1$. Prove $A^n=0_n$. ...
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3answers
66 views

Evaluate a $3\times3$ determinant. [closed]

Show that $$\left|\begin{matrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{matrix}\right|=(a-b)(b-c)(c-a)$$
3
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1answer
30 views

Determinant of hankel matrix of hyperbolic functions, $a_n=\frac{n}{\sinh(\pi n)}$

I am trying to learn about the properties of Hankel matrices, and they appear to have nice closed forms for quite a large class of sequences. The class I am interested is when the elements $a_n$ are ...
1
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1answer
57 views

If the determinant of a matrix goes to infinity, does it means it has no inverse?

Context I have a linear time-invariant (single-input, single-output) system in state space representation (https://en.wikipedia.org/wiki/State-space_representation#Linear_systems): $$ \mathbf{x'}(t) ...
0
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1answer
21 views

Rule of thumb on number of zero entries for invertibility of a $4\times 4 $ matrix?

I have to determine whether a $4\times 4$ matrix $A$ is invertible. Suppose that there are no zero columns or zero rows. Is there any rule of thumb saying how many zero entries can be at most in $A$, ...
0
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1answer
62 views

How prove this determinant can't zero

Let $x,y,z\neq 0$ be real numbers, show that $$f(x,y,z)=\begin{vmatrix} \sqrt{x^2+y^2}&|x|&|y|\\ |y|&\sqrt{y^2+z^2}&|z|\\ |x|&|z|&\sqrt{x^2+z^2} \end{vmatrix}\neq 0$$ or it ...
1
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1answer
52 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times n}'...
2
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2answers
35 views

How do I find value of a and b in this matrix question?

This is a question from a homework sheet my teacher gave. I already did alternate a. Alternate b is quite confusing! It asks to find the value for a and b. I don't really know what to do but here's ...
0
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1answer
52 views

Calculation of characteristic polynomial

I have to determine the characteristic polynomial of the matrix $$A = \begin{pmatrix} 0 & 0 &\cdots &0& -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots &...
4
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1answer
88 views

Calculating the determinant of a matrix using its rank

Let A, B, C and D be real n×n matrices. If $$\operatorname{rank} \begin{bmatrix} \ A & B \\[0.3em] \ C & D \\[0.3em] \end{bmatrix} = n$$ then show that $$\det \begin{...
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1answer
54 views

Adjoint of an adjoint of a matrix

Can you please help me on this question? $\DeclareMathOperator{\adj}{adj}$ $A$ is a real $n \times n$ matrix; show that: $\adj(\adj(A)) = (\det A)^{n-2}A$ I don't know which of the expressions ...
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0answers
20 views

Inverse Gramian matrix

Show that inverse Gramian matrix is Gramian matrix Here is my idea. $\Gamma ^ {-1} = \frac{A}{|\Gamma|}$, where A is transposed matrix of cofactors (not sure about the term, correct me please), |Г| ...
4
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1answer
33 views

Matrix consisting of cosines of differences

Consider the following matrix: $$\left[\begin{array}{cccc} \cos(x_1-y_1) & \cos(x_1-y_2) & \ldots & \cos(x_1-y_n) \\ \cos(x_2-y_1) & \cos(x_2-y_2) & \ldots & \cos(x_2-y_n) \\ ...
3
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3answers
41 views

Matrix with a certain pattern

Consider the following matrix: $$\left[\begin{array}{cccc} 1+x_1y_1 & 1+x_1y_2 & \ldots & 1+x_1y_n \\ 1+x_2y_1 & 1+x_2y_2 & \ldots & 1+x_2y_n \\ 1+x_3y_1 & 1+x_3y_2 & ...
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1answer
18 views

For integers $n>1$ , $k$ , does there exist matrix $A$ with integer entries and first row $(1,2,…,n)$ such that $\det A=k$?

Let $n >1$ be an integer , then is it true that for any integer $k$ , there exist a matrix $A \in M(n,\mathbb Z)$ with first row of $A$ as $(1,2,...,n)$ such that $\det A=k$ ?
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0answers
106 views

Prove that $ \det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$ [duplicate]

Let $A,B \in M_n(\mathbb{R})$. Prove that $\det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$. I know that there is a theorem which says that if $E,F,G,H \in M_n(\mathbb{F})$ and if $...
7
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2answers
273 views

Non-negative determinant of a block matrix

Here's the problem I've been stuck on for some time now. Let $A,B \in M_n(\mathbb{R})$. Let $C= \begin{bmatrix} A & B \\ -B & A \\ \end{bmatrix} $ be a real ...
4
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1answer
60 views

Inverting an $n \times n$ matrix using determinant

We're asked to invert the following matrix with the help of guided questions. $$\begin{pmatrix} 1 + a_1 & 1 & \cdots & 1 \\ 1 & 1+a_2 & \ddots & \vdots \\ \vdots & \ddots &...
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1answer
91 views

Determinant of augmented matrices.

Let $A$ and $B$ be $n \times n$ real matrices. How can I show that $\det \begin{bmatrix} A & B \\[0.3em] -B & A \\[0.3em] \end{bmatrix} \geq 0 $?
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1answer
45 views

Derivatives using matrices good

$$\left|\begin{matrix} (1+x)^{a_1b_1} & (1+x)^{a_1b_2} & (1+x)^{a_1b_3} \\ (1+x)^{a_2b_1} & (1+x)^{a_2b_2} & (1+x)^{a_2b_3} \\ (1+x)^{a_3b_1} & (1+x)^{a_3b_2} & (1+x)^{a_3b_3} \...
1
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1answer
33 views

The property that det(A) = prod of A's eigenvalues, and tr(A) = sum of A's eigenvalues

Do these two properties fail to be true, if A's characteristic polynomial fails to split? If so, then do we usually work in a vector space with the ground field = $\mathbb{C}$, when we want to use ...
2
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0answers
27 views

Show that matrix is totally unimodular

I want to show that this matrix is totally unimodular: \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & ...
4
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2answers
53 views

When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A = $$ \begin{bmatrix} 2 & 0 &0\\ ...
0
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0answers
18 views

Coefficient of bivariate polynomial as a determinant of matrix

Given $$ \begin{bmatrix} a\\ b\\ c\\ d\\ \end{bmatrix}=\begin{bmatrix} a_0t^3+a_1st^2+a_2s^2t+a_3s^3\\ a_4t^2+a_5st+a_6s^2\\ a_7t+a_8s\\ a_9\\ \end{bmatrix} $$ the following equation holds: $$ -\...
2
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1answer
64 views

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ \lambda_1^{p_3}...
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0answers
50 views

Given matrices $A,B, \det(A) = 2$ find $\det(B)$

Given matrices $A$,$B$, $\det(A) = 2$ Find $\det(B)$ $$A= \begin{pmatrix} a & 1 & b \\ 2 & 3a & 1 \\ b & 1 & 2a \\ \end{pmatrix}, \...
7
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1answer
624 views

Determinant of Matrix with uncomputable values.

Calculate the determinant of the matrix $$ \begin{pmatrix} 10^{10} & 10^{10^{10}} & 11^{11^{11}} & 1 & 0 \\ 2^{2^2} & 3^{3^3} & 7^{7^7} & 0 & 1 \\ 11 & 17 &...
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2answers
28 views

Proving $\det \big(Df\big|_x\big)=0$ for a function into unit circle

Let $f:\mathbb{R}^2\to S$ where $S=\{x\in\mathbb{R}^2:\, ||x||=1\}$. Prove that $\det \big(Df\big|_x\big)=0$ for all $x$. I'm having trouble attacking this. So I need to show that there is some $\...
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0answers
10 views

Proof attempt: A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B| [duplicate]

I asked this question but all the answers I got were outside of my scope of understanding, so here is as close as I got to a solution: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ ...
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1answer
24 views

tell Positive Definite Matrices by the sign of determinants

This is from my textbook I don't understand why it didn't mention the other situation which is $det(A_k) <0 $ for all k, and we stall have positive pivot because $\frac{negative}{negative}=...
3
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3answers
112 views

Prove that $\det(A^{T}A) \neq 0$

How to prove that $\det(A^{T}A) \neq 0$ if coloumns of $A$ are linearly independent, without using Cauchy-Binet formula? $A$ is real matrix.
1
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1answer
60 views

Finding determinant of matrix through row operations [problem help]?

I am having trouble understanding a problem that my Linear Algebra class gave. I understand that determinants can be found through row operations with the following points: 1.) Adding a multiple ...
0
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0answers
42 views

determinant of a matrix with increased off-diagonal elements

I have symmetric matrix which is formed by complex integer vectors as follows \begin{bmatrix} \|f_1\|^2 & af_2^Hf_1 & bf_3^Hf_1 & \dots \\ a^*f_1^Hf_2 & \|f_2\|^2 & cf_3^Hf_2 & ...
2
votes
0answers
49 views

Determinants Proof Can't Solve

Let $v_1,\cdots ,v_n$ be vectors in $\mathbb R^n$. Define $w_i$ as the vector in $\mathbb R^{n+1}$, which is just $v_i$ with a $0$ added to the beginning. (So if $v_i = (1,0,1)$, then $w_i = (0,1,0,...
-7
votes
1answer
120 views

Generalized determinant of order $n+2$

How to solve following determinant of order $n+2$ to get eigenvalues? $\begin{vmatrix} -\lambda & 0 & 1 & 1 & 1 & \cdots & 1 \\ \dfrac{n}{\lambda} & -\lambda+\dfrac{n}{\...
0
votes
1answer
22 views

A family of vectors is linearly independent.

Let $K$ be a field and $E$ be a $K$-vector space of dimension $n$. Let $\phi$ be an endomorphism of $E$. Let $(\lambda_1,\cdots,\lambda_n)$ be a family of distinct scalars and $(x_1,\cdots,x_n)$ be ...
3
votes
0answers
21 views

Determinant of $\delta$ function

Let $$\delta_i^j=\left\{ \begin{aligned} 1 ~~~~~~i=j \\ 0 ~~~~~~i\ne j \end{aligned} \right. $$ $1\le i,j\le n$. How to prove $$ \begin{vmatrix} \delta_{j_1}^{i_1} ~...~ \delta_{j_n}^{i_1} \\ \\ \...
0
votes
1answer
17 views

Show that every curvature of a Frenet curve satisfy the following statement.

I need to show the following statement: Show that for every Frenet curve $c:I\to\mathbb{R}^n$, the curvatures $\kappa_1(t),\ldots,\kappa_{n-1}(t)$ satisfy the following equality: $$\prod_{i=1}^{n-1}(...