3
votes
1answer
33 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 ...
0
votes
1answer
23 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
79 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), $$
just curious.
0
votes
2answers
32 views
Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices
Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
-1
votes
3answers
113 views
Evaluation of a specific determinant.
Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$.
$$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
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
$$\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le ...
3
votes
1answer
79 views
Skew symmetric matrix decomposes
I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $ R $ such that $ R^T A R = M$, ...
0
votes
1answer
50 views
Linear algebra determinants
I have tried to solve this problem but I don't have an idea how to begin, any hints?
For any vector $x$ in $\mathbb{R}^n$ let $(x,x) =\sum\limits_{i=1}^n x_i^2 $ . Let $A$ be a matrix of size $n ...
1
vote
1answer
65 views
prove that determinant is a quadratic form
let $V$ be a vector space of all $2 \times 2$ hermitian matrices with entries from $\mathbb C$, over the field $\mathbb R$.
prove that $q(v)=\det(v)$ is a quadratic form.
I tried to prove that ...
2
votes
1answer
70 views
Determinant problem
I'm stuck in this question:
How calculate this determinant ?
$$\Delta=\left|\begin{array}{cccccc}
1&2&3&\cdots&\cdots&n\\
n&1&2&\cdots&\cdots& n-1\\
...
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 ...
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 ...
2
votes
1answer
44 views
Solving linear equations with Vandermonde
Given this:
$$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & ...
2
votes
1answer
105 views
Determinant of matrix?
How can we calculate the determinant of this $\,pn\times pn\,$ matrix. I have tried at my best level, and still am not able to come up with a solution. The matrix $a_{ij}$ entry is defined as
$$
...
0
votes
4answers
38 views
Divide and Conquer matrices to calculate determinant.
Do the determinant of a matrix equal to the determinant of submatrices?
$$
det\begin{pmatrix}
a_{11} & a_{12} & a_{13} & \dots & a_{1k} \\
a_{21} & a_{22} & a_{23} & ...
1
vote
1answer
45 views
a problem on solving a determinant equation [duplicate]
Let $a$ be a real number. What is the number of distinct real roots of the following
$$\left| \begin{array}{ccc}
x & a & a & a \\
a & x & a & a \\
a & a & x & a \\
...
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 ...
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.
$$D_n= \det \begin{pmatrix}
...
1
vote
1answer
77 views
How to show by induction that, for $0<\theta<\pi$, $\det A_n=\frac{\sin (n+1)\theta}{\sin \theta}.$
I need help with the underlined part.
Thanks in advance
Let $A_n$ be the $n\times n$ matrix given by
$$a_{ij}=
\begin{cases}
0 & \text{if }|i-j|>1, \\
1 & \text{if }|i-j|=1, ...
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 ...
2
votes
3answers
80 views
Prove that $\operatorname{adj}A^t = \operatorname{adj} A$
Let $A$ be an anti-symmetric ($A^t = -A$), squared matrix ($n \times n$, while $n$ is uneven).
Prove that ${\rm adj}\;A^t = {\rm adj}\;A$.
1
vote
1answer
31 views
If $f(X) = a_0 + a_1 X + a_2 X^2 \in \mathbb{F}[X]$ then show $f$ is uniquely determined by $f(x)$, $f(y)$, $f(z)$?
This is the exact question:
It's part(ii) that I don't understand - what does it mean and what is it asking me to do? How would I go about constructing a proof? Any help would be much appreciated.
0
votes
0answers
35 views
Determinant of a matrix whose diagonal entries are x and the other entries are 1 [duplicate]
Suppose $A$ is an $n$-by-$n$ matrix. Its diagonal entries are $x$ and the other entries are $1$. Find the determinant of $A$.
I got $\det(A)=(x+n-1)^n*(x-1)^{(n-1)}$
Is that the correct answer?
2
votes
5answers
103 views
Calculation of $\lambda$ in determinant multiplication.
$$\begin{vmatrix}
a^2+\lambda^2 & ab+c\lambda & ca-b\lambda \\
ab-c\lambda & b^2+\lambda^2& bc+a\lambda\\
ca+b\lambda & bc-a\lambda & c^2+\lambda^2
...
4
votes
2answers
51 views
$A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$
I found the following problem interesting but do not know how to tackle it.
If $A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$.Then find $\det (A^2+B^2)=?.$
Can ...
1
vote
1answer
54 views
Question on Hoffman and Kunze's proof of the Cayley-Hamilton theorem: why is $ \det (xI-A) =x^2-\mathrm{Tr}(A)*x+\det(A) $
At one point, in the proof of the Cayley-Hamilton theorem the authors say that $$\det (xI-A) =x^2-\mathrm{Tr}(A)*x+\det(A)$$ for any $n\times n$ matrix that represents a linear operator, $I$ being the ...
8
votes
3answers
249 views
Is $\;\det(A^n) =\left(\det (A)\right)^n\;$?
How can the value of $\;\det\left(A^{11}\right)\;$ be calculated from $\;\det(A)$?
Generally how can $\;\det\left(A^n\right)\;$ be obtained from $\;\det(A)$?
0
votes
2answers
49 views
Determinants and diagonalizability
Does the determinant of a matrix affect if it is diagonalizable or not?
Like, if $\det(A) = 0$ does that mean the matrix is NOT diagonalizable?
3
votes
1answer
60 views
Fast way to calculate determinant for a block matrix
I have a block matrix
$$Q_{(n+m-1)\times(n+m-1)} = \begin{pmatrix} A & -J\\-J^t & B \end{pmatrix}$$
where
$$A_{(m-1)\times(m-1)} = n*I_{(m-1)\times(m-1)} \text{ and } B_{n\times n} = ...
2
votes
5answers
203 views
what does it mean to have a determinant equal to zero
After looking in my book for a couple of hours I'm still confused what does it mean to have a determinant equal to zero, I hope someone can explain me in plain English. many thanks
1
vote
2answers
64 views
Matrix Identity Proof
Let $A$ and $C$ be $3 \times 2$ matrices and let $B$ be a $2 \times 2$ matrix such that $AB=C$. Prove that:
$$||A_1 \times A_2 || \cdot |\det B| = ||C_1 \times C_2 ||$$
where $A_i$ and $C_i$ are the ...
5
votes
3answers
96 views
Special orthogonal matrices have orthogonal square roots
Let $A$ be an orthogonal matrix with $\det (A)=1$. Show that there exists an orthogonal matrix $B$ such that $B^2=A$.
Thank you very much.
0
votes
2answers
69 views
Linear algebra: need help with proof
Can someone please help me with this proof.
For $A,B$ ∈ $F^{n×n}$, show that $AB$ and $BA$ have the same characteristic polynomial.
2
votes
1answer
30 views
Finding Determinants Recursively
From the MIT OCW Linear Algebra (18.06) final exam, question 9:
For square matrices with 3's on the diagonal, 2s on the diagonal
above, and 1s on the diagonal below:
$$A_1=\begin{pmatrix} 3 ...
1
vote
2answers
66 views
Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$
Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$
Ran into trouble with a proof for linear algebra. $C$ is the cofactor matrix of $A \in \mathbb{R}^{n\times n}$, and I'm not sure how to ...
0
votes
1answer
37 views
Determinant formula and invertibility.
I am working on a problem where I need to find the determinant of
$$
\begin{bmatrix}
b & a & & \\
& b & a \\
& & & \ddots \\
& & & & ...
