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.
3
votes
1answer
34 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 ...
3
votes
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 ...
1
vote
1answer
32 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 ...
0
votes
1answer
24 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?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
0
votes
3answers
26 views
$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form
Prove that every $n$-linear alternating form on a vector space of dimension
less than $n$ is the zero form.
4
votes
1answer
69 views
When does a matrix $A$ with ones on and above the diagonal have $\det(A)=1$?
What conditions, if they're even necessary, must be placed on $\star$ so that the matrix
$$ \begin{pmatrix} 1 & & \huge{1} \\ & \ddots & \\ \huge{\star} & & 1 \end{pmatrix}, ...
1
vote
2answers
84 views
+100
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 $u$.
$$
\int_{\triangle}|\nabla ...
4
votes
2answers
78 views
Determinants: A Special Condition
Under what conditions is
$$ \det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n), $$
just curious.
5
votes
3answers
68 views
How to show that $\det(A+I)\ne 0$
How to show that for any skew symmetric real matrix $A$, we have $\det(A+I)\ne 0?$
Where to begin? I'm looking for some clue only.
0
votes
2answers
33 views
Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices
Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
-1
votes
3answers
113 views
Evaluation of a specific determinant.
Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$.
$$A_2
\begin{pmatrix} 1& 1\\
1& 2
\end{pmatrix};
A_3 = ...
1
vote
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 $d_n(\sqrt{2})$.
...
4
votes
2answers
114 views
Multiplication of determinants
Show that for any vectors $\bf{a}$,$\bf{b}$,$\bf{c}$,$\bf{u}$,$\bf{v}$,$\bf{w}$ in $\mathbb{R}^3$,
...
1
vote
2answers
45 views
How to calculate the determinant of a matrix using Laplace?
How to calculate the determinant using Laplace?
$$
\det \begin{bmatrix}
0 & \dots & 0 & 0 & a_{1n} \\[0.3em]
0 & \dots & 0 & a_{2,n-1} & ...
1
vote
2answers
82 views
Simplest way to calculate a determinant [duplicate]
The big $1$'s here just mean that the lower and upper triangular entries are all $1$'s. The trace entries are all zero. The matrix is for a general $n\times n$ matrix of this form. I'm trying to ...
3
votes
0answers
32 views
What is the limit $\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le m/n\le r/s}x^my^n$?
Let $S=[0,1)^2$ and $m,n$ are positive integers and $p/q,r/s$ are positive rationals with $p/q<r/s$. What is the limit
$$\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le ...
3
votes
1answer
79 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$, ...
0
votes
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 ...
1
vote
1answer
67 views
How to calculate the determinant of a matrix with …
How to calculate the determinant using Laplace?
$$
\det \begin{bmatrix}
-t & 0 & 0 & \dots & 0 & a_1 \\[0.3em]
a_2 & -t & 0 & \dots & 0 ...
1
vote
1answer
65 views
prove that determinant is a quadratic form
let $V$ be a vector space of all $2 \times 2$ hermitian matrices with entries from $\mathbb C$, over the field $\mathbb R$.
prove that $q(v)=\det(v)$ is a quadratic form.
I tried to prove that ...
2
votes
1answer
70 views
Determinant problem
I'm stuck in this question:
How calculate this determinant ?
$$\Delta=\left|\begin{array}{cccccc}
1&2&3&\cdots&\cdots&n\\
n&1&2&\cdots&\cdots& n-1\\
...
1
vote
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 ...
2
votes
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} \\
... & ... & ...
0
votes
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 ...
-1
votes
3answers
40 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:
$$
M = ...
4
votes
2answers
44 views
Calculate the determinant when the sum of odd rows $=$ the sum of even rows
I have came across this interesting question in linear algebra and I couldn't know for sure the answer.
Given a matrix $A \in M_{n \times k} (\mathbb F)$, The sum of odd rows of $A$ $=$ the sum of ...
0
votes
0answers
47 views
Generalizing formula for calculating determinant of specific matrix
There is a similar question like this. And this is extension of this question
How can we calculate the determinant of this $\,pn-1\times pn-1\,$ matrix. I have tried at my best level, and still am ...
0
votes
2answers
107 views
Computing determinant of this matrix
I have a very specific kind of matrix and I have to find the formula to find the determinant of these matrix.
a(i,j)=a if(i==j) and a(i,j)=0 if(floor(i/2)=floor(j/2) and i!=j) and n is odd
$$ ...
2
votes
1answer
44 views
Solving linear equations with Vandermonde
Given this:
$$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & ...
12
votes
3answers
194 views
Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$
I have to prove/disprove this:
If $\det(A+X) = \det(B + X)~ \forall X \in M_{n \times n} (\mathbb F) \rightarrow A = B$
I believe it is true but I can not think of a direct way to prove it.
Any ...
2
votes
2answers
55 views
Calculating the determinant of this matrix
Given this (very) tricky determinant, how can we calculate it easily?
$$\begin{pmatrix} \alpha + \beta & \alpha \beta & 0 & ... & ... & 0 \\ 1 & \alpha + \beta & \alpha ...
2
votes
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
$$
...
0
votes
4answers
38 views
Divide and Conquer matrices to calculate determinant.
Do the determinant of a matrix equal to the determinant of submatrices?
$$
det\begin{pmatrix}
a_{11} & a_{12} & a_{13} & \dots & a_{1k} \\
a_{21} & a_{22} & a_{23} & ...
1
vote
1answer
46 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
$$\left| \begin{array}{ccc}
x & a & a & a \\
a & x & a & a \\
a & a & x & a \\
...
2
votes
1answer
51 views
Simple/Concise proof of Muir's Identity
I am not a Math student and I am having trouble finding some small proof for the Muir's identity.
Even a slightly lengthy but easy to understand proof would be helpful.
Muir's Identity
$$\det(A)= ...
3
votes
1answer
90 views
Different form of determinant, does it make mine wrong?
Calculate the determinant of the following $(n+1) \times (n+1)$ matrix:
$$A = \pmatrix{1 & 1 & 1 & 1 &\cdots & 1 \\ 1 & a_1 & 0 & 0 &\cdots & 0 \\ 1 ...
1
vote
1answer
41 views
Determinant is correct but wrong when I try and check it
I have to work out the determinant of the $(n \times n)$ matrix
$$A = \pmatrix{x & y & 0 & 0 &\cdots & 0 \\ 0 & x & y & 0 &\cdots & 0 \\ 0 & 0 & x ...
1
vote
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? ...
1
vote
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 ...
4
votes
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?
3
votes
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.
$$D_n= \det \begin{pmatrix}
...
0
votes
1answer
17 views
show that$v(E) = a_1a_2a_3…a_nv(B^n)$
I'm generally pretty good a change in variable type problems, but this one has me stumped. It's on page 264 in Advanced calculus of several variables by Edwards.
Thm 5.1: If $\lambda:R^n \rightarrow ...
1
vote
1answer
77 views
How to show by induction that, for $0<\theta<\pi$, $\det A_n=\frac{\sin (n+1)\theta}{\sin \theta}.$
I need help with the underlined part.
Thanks in advance
Let $A_n$ be the $n\times n$ matrix given by
$$a_{ij}=
\begin{cases}
0 & \text{if }|i-j|>1, \\
1 & \text{if }|i-j|=1, ...
3
votes
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 ...
0
votes
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 ...
2
votes
1answer
86 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 ...
2
votes
1answer
86 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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
