# Tagged Questions

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.

1answer
28 views

### Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$

Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$ Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it. Thanks for ...
2answers
27 views

### Linear Algebra determinant and rank relation

True or False? If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$. Is it false or true? My guess is true, because the matrix $A$ is invertible. But there is ...
1answer
27 views

### Computing Resultant

The resultant of two polynomials is defined as the determinant of the Sylvester matrix. If the polynomials are of degree $n$ and $m$, than the Sylvester matrix will be of dimension \$(m+n)\times ...
1answer
21 views

### Column entries of a matrix sum to zero, so what are the properties?

What kind of properties does a matrix whose column entries sum to zero have?  \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & ...
3answers
26 views

### $n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form

Prove that every $n$-linear alternating form on a vector space of dimension less than $n$ is the zero form.
1answer
69 views

### When does a matrix $A$ with ones on and above the diagonal have $\det(A)=1$?

What conditions, if they're even necessary, must be placed on $\star$ so that the matrix  \begin{pmatrix} 1 & & \huge{1} \\ & \ddots & \\ \huge{\star} & & 1 \end{pmatrix}, ...
0answers
33 views

### On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$.  \int_{\triangle}|\nabla ...
2answers
77 views

### Determinants: A Special Condition

Under what conditions is $$\det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n),$$ just curious.
3answers
68 views

### How to show that $\det(A+I)\ne 0$

How to show that for any skew symmetric real matrix $A$, we have $\det(A+I)\ne 0?$ Where to begin? I'm looking for some clue only.
2answers
31 views

### Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices

Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
3answers
112 views

### Evaluation of a specific determinant.

Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$. A_2 \begin{pmatrix} 1& 1\\ 1& 2 \end{pmatrix}; A_3 = ...
0answers
35 views

### Determinant of a matrix with variables in it

Assuming that $z \neq 0$, compute the determinant $d_n(z) = \det D_n \left(1, z, 1 - \frac{1}{z^2} \right)$, where $z$ is a complex variable. In particular, compute the value $d_n(\sqrt{2})$. ...
2answers
114 views

### Multiplication of determinants

Show that for any vectors $\bf{a}$,$\bf{b}$,$\bf{c}$,$\bf{u}$,$\bf{v}$,$\bf{w}$ in $\mathbb{R}^3$, ...
2answers
45 views

### How to calculate the determinant of a matrix using Laplace?

How to calculate the determinant using Laplace?  \det \begin{bmatrix} 0 & \dots & 0 & 0 & a_{1n} \\[0.3em] 0 & \dots & 0 & a_{2,n-1} & ...
2answers
81 views

### Simplest way to calculate a determinant [duplicate]

The big $1$'s here just mean that the lower and upper triangular entries are all $1$'s. The trace entries are all zero. The matrix is for a general $n\times n$ matrix of this form. I'm trying to ...
0answers
32 views

### What is the limit $\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le m/n\le r/s}x^my^n$?

Let $S=[0,1)^2$ and $m,n$ are positive integers and $p/q,r/s$ are positive rationals with $p/q<r/s$. What is the limit \lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le ...
1answer
77 views

### Skew symmetric matrix decomposes

I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $R$ such that $R^T A R = M$, ...
1answer
50 views

### Linear algebra determinants

I have tried to solve this problem but I don't have an idea how to begin, any hints? For any vector $x$ in $\mathbb{R}^n$ let $(x,x) =\sum\limits_{i=1}^n x_i^2$ . Let $A$ be a matrix of size \$n ...
1answer
67 views

### How to calculate the determinant of a matrix with …

How to calculate the determinant using Laplace?  \det \begin{bmatrix} -t & 0 & 0 & \dots & 0 & a_1 \\[0.3em] a_2 & -t & 0 & \dots & 0 ...
1answer
65 views

### prove that determinant is a quadratic form

let $V$ be a vector space of all $2 \times 2$ hermitian matrices with entries from $\mathbb C$, over the field $\mathbb R$. prove that $q(v)=\det(v)$ is a quadratic form. I tried to prove that ...
1answer
70 views

### Determinant problem

I'm stuck in this question: How calculate this determinant ? \Delta=\left|\begin{array}{cccccc} 1&2&3&\cdots&\cdots&n\\ n&1&2&\cdots&\cdots& n-1\\ ...
2answers
50 views

### Product of two matrices equals zero

If the product of two $n \times n$ matrices $A$ and $B$ is zero ie: $AB = 0$ Then either $\det(A)$ or $\det(B)$ must be zero. What additional conditions on $A$ and $B$ would be sufficient ? Clearly ...
2answers
43 views

### Characteristic and minimal polynomial of a special matrix

\$H = \begin{bmatrix} 1 & w^{-1} & w^{-2} & ... & w^{1-n}\\ w & 1 & w^{-1} & ... & w^{2-n} \\ w^{2} & w^1 & 1 & ... & w^{3-n} \\ ... & ... & ...
3answers
51 views

### Characteristic value or eigenvalues and determinant

I am having semester in linear algebra. And have recently got acquainted to eigenvalues. What is the relation between eigenvalues and determinant? Going through answers of some questions I found ...
3answers
39 views

### Questions about matrices and determinants - constant variable multiplication

Is this matrix $$M = \begin{bmatrix} a & -a & a \\[0.3em] -a & -a & -a \\[0.3em] a & a & a \end{bmatrix}$$ the same as:  M = ...
2answers
44 views

### Calculate the determinant when the sum of odd rows $=$ the sum of even rows

I have came across this interesting question in linear algebra and I couldn't know for sure the answer. Given a matrix $A \in M_{n \times k} (\mathbb F)$, The sum of odd rows of $A$ $=$ the sum of ...
0answers
46 views

### Generalizing formula for calculating determinant of specific matrix

There is a similar question like this. And this is extension of this question How can we calculate the determinant of this $\,pn-1\times pn-1\,$ matrix. I have tried at my best level, and still am ...
2answers
107 views

### Computing determinant of this matrix

I have a very specific kind of matrix and I have to find the formula to find the determinant of these matrix. a(i,j)=a if(i==j) and a(i,j)=0 if(floor(i/2)=floor(j/2) and i!=j) and n is odd  ...
1answer
44 views

### Solving linear equations with Vandermonde

Given this: \begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & ...
3answers
192 views

### Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$

I have to prove/disprove this: If $\det(A+X) = \det(B + X)~ \forall X \in M_{n \times n} (\mathbb F) \rightarrow A = B$ I believe it is true but I can not think of a direct way to prove it. Any ...
2answers
53 views

### Calculating the determinant of this matrix

Given this (very) tricky determinant, how can we calculate it easily? \begin{pmatrix} \alpha + \beta & \alpha \beta & 0 & ... & ... & 0 \\ 1 & \alpha + \beta & \alpha ...
1answer
105 views

### Determinant of matrix?

How can we calculate the determinant of this $\,pn\times pn\,$ matrix. I have tried at my best level, and still am not able to come up with a solution. The matrix $a_{ij}$ entry is defined as  ...
4answers
38 views

### Divide and Conquer matrices to calculate determinant.

Do the determinant of a matrix equal to the determinant of submatrices?  det\begin{pmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1k} \\ a_{21} & a_{22} & a_{23} & ...
1answer
45 views

### a problem on solving a determinant equation [duplicate]

Let $a$ be a real number. What is the number of distinct real roots of the following \left| \begin{array}{ccc} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ ...
1answer
51 views

### Simple/Concise proof of Muir's Identity

I am not a Math student and I am having trouble finding some small proof for the Muir's identity. Even a slightly lengthy but easy to understand proof would be helpful. Muir's Identity \det(A)= ...
1answer
90 views

### Different form of determinant, does it make mine wrong?

Calculate the determinant of the following $(n+1) \times (n+1)$ matrix: A = \pmatrix{1 & 1 & 1 & 1 &\cdots & 1 \\ 1 & a_1 & 0 & 0 &\cdots & 0 \\ 1 ...
1answer
41 views

### Determinant is correct but wrong when I try and check it

I have to work out the determinant of the $(n \times n)$ matrix A = \pmatrix{x & y & 0 & 0 &\cdots & 0 \\ 0 & x & y & 0 &\cdots & 0 \\ 0 & 0 & x ...
1answer
26 views

### Problem related to a complex matrix

I am stuck on the following problem: Let $P$ be a $2 \times 2$ complex matrix such that trace $P=1$ and $\det P=-6.$ Then trace $(P^4-P^3)=?$ Can someone point me in the right direction? ...
1answer
23 views

### Definition of minimal and characteristic polynomials

I have defined the characteristic and minimal polynomial as follows, but have been told this is not strictly correct since det$(-I)$ is not necessarily 1, so my formulae don't match for $A=0$, how can ...
2answers
78 views

### How can I prove $\det(\overline M)=\overline{\det(M)}$?

Of course $\overline M$ is the complex conjugate of an $n\times n$ matrix $M$. Someone gave me advice to use the definition of determinant, then it means I have to use cofactor expasion here?
2answers
58 views

### Is this determinant bounded?

Let $D_n$ be the determinant of the $n-1$ by $n-1$ matrix such that the main diagonal entries are $3,4,5,\cdots,n+1$ and other entries being $1$. i.e. D_n= \det \begin{pmatrix} ...
1answer
17 views

### show that$v(E) = a_1a_2a_3…a_nv(B^n)$

I'm generally pretty good a change in variable type problems, but this one has me stumped. It's on page 264 in Advanced calculus of several variables by Edwards. Thm 5.1: If \$\lambda:R^n \rightarrow ...
1answer
77 views

### How to show by induction that, for $0<\theta<\pi$, $\det A_n=\frac{\sin (n+1)\theta}{\sin \theta}.$

I need help with the underlined part. Thanks in advance Let $A_n$ be the $n\times n$ matrix given by a_{ij}= \begin{cases} 0 & \text{if }|i-j|>1, \\ 1 & \text{if }|i-j|=1, ...
1answer
49 views

### Find the smallest square matrix in which some objects fit following some rules

I have to put some objects in a matrix. The data of these objects is given in another matrix in which each line contains an object, and the first column represents its width, and the second its ...
0answers
18 views

### Hyperdeterminants

Beyond the Cayley hyperdeterminant, are there an explicite general rule to calculate hyperdeterminants for any hypermatrix? I mean a rule to calculate the hyperdeterminant with format \$k_1\times ...
1answer
84 views

### What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?

Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this ...
1answer
84 views

### Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory

In order to proof the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes de product of ...
1answer
57 views

### Problem with Jacobi's formula for determinants

Jacobi's formula says that: $$\det e^{X}=e^{\operatorname{Tr}(X)}$$ So for any matrix $A$, I could try to find a matrix $X$ (the equivalent to a group generator) such that $A=e^{X}$ holds. But if ...
1answer
25 views

### proof about deteminant of a complex linear transformation

say I have a linear space $V$ over $\Bbb C$ and a linear transformation $T:V \to V$ such that $T=A+iB$ where $A,B \in \Bbb R^{n \times n}$ I proved already that \$T_\Bbb R = \begin{pmatrix} A & -B ...
2answers
56 views

### deteminant of a block skew-symmetric matrix

If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix} how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix} so the determinant is ...
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