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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5
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152 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
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2answers
103 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
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2answers
121 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 ...
7
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1answer
117 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
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1answer
125 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
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1answer
52 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
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1answer
83 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
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1answer
161 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
2answers
236 views

Circular 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
458 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 ...
3
votes
2answers
75 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
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3answers
422 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
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3answers
75 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
92 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
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0answers
67 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
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2answers
124 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
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1answer
157 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
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3answers
514 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
79 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
160 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
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4answers
211 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
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1answer
64 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
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1answer
79 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
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1answer
106 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 ...
2
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1answer
73 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 ...
0
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1answer
50 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? ...
0
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1answer
86 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
1
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1answer
67 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
132 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
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2answers
83 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
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1answer
19 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
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1answer
132 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
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1answer
93 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 ...
2
votes
1answer
1k 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 ...
4
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2answers
4k views

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

In order to prove 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 thr product of ...
3
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1answer
211 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
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1answer
76 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
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2answers
213 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
137 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
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1answer
69 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.
3
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2answers
95 views

How to solve this determinant?

I have to solve determinant of the following form: $$a_{ij}=|i-j|+1$$ It looks like this: $$ \begin{pmatrix} 1 & 2 & 3 & 4 & \cdots & n \\ 2 & 1 & 2 & 3 & ...
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2answers
622 views

Trace of the matrix power

Say I have matrix $A = \begin{bmatrix} a & 0 & -c\\ 0 & b & 0\\ -c & 0 & a \end{bmatrix}$. What is matrix trace tr(A^200) Thanks much!
2
votes
5answers
196 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 ...
5
votes
5answers
381 views

How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix $$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$ How to prove that ...
4
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2answers
102 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 ...
2
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1answer
113 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 ...
0
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1answer
69 views

Calculate $|X|$ , where $X=(A+A^2B^2+A^3+A^4B^4\dots _{100 \ terms})$

$$A = \left[ \begin{array}{rrr} 2 & -2 & -4 \\\ -1 & 3 & 4 \\\ 1 & -2 & -3 \end{array} \right]$$ $$B = \left[ \begin{array}{rrr} -4 & -3 & -3 \\\ 1 ...
7
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3answers
284 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 $\;\det(A)$?
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2answers
891 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
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6answers
191 views

Matrix inverse identity

Question: Assuming that all matrix inverses involved below exist, show that $$(\mathbf{A}-\mathbf{B})^{-1}=\mathbf{A}^{-1}+\mathbf{A}^{-1}(\mathbf{B}^{-1}-\mathbf{A}^{-1})^{-1}\mathbf{A}^{-1}$$ in ...