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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0
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1answer
25 views

A different determinant expansion

Let us denote the determinant of a matrix $A=((a_{ij}))_{i,j=1}^n$ as $\displaystyle \det_{i,j=1}^n a_{ij}$. Let $\delta_{ij}$ be the kronecker delta function and suppose $a_{ij}=a_{ji}$ for ...
2
votes
2answers
40 views

Compute the indicated power of a matrix

Compute the indicated power of the matrix: $A^8$ $ A = \begin{bmatrix}2&1&2\\2&1&2\\2&1&2\end{bmatrix} $ I calculated the eigenvalues: $ \lambda_1 = \lambda_2 = 0, \lambda_3 ...
-3
votes
1answer
46 views

$A$ and $B$ are $3\times3$ matrices. $3A-B^2=0$ and $A^2-4B=0$, find possible determinants for A and B

$$3A-B^2=0$$ $$A^2-4B=0$$ I don't know how to solve that. I've tried solving for some general $3a_{ij}$ and express the element that is deducted but it doesn't get me anywhere...
0
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1answer
16 views

Directional derivative of determinant at the identity is the trace of the matrix?

Let $f:A\mapsto \rm{det}(A)$, Prove that $\left(Df\right)_{{\rm id}}\left(H\right)={\rm tr}\left(H\right)$ for all $H\in\mathcal{L}\left(\mathbb{R}^{n}\to\mathbb{R}^{n}\right)$. The question ...
8
votes
1answer
140 views

Determinant of $n\times n$ matrix with parameter

Problem: Let $\delta \in \mathbb{R}^+$ and $n\in \mathbb{N}$. The matrix $A_n = (a_{i,j}) \in \mathbb{R}^{n\times n}$ is defined as $$ a_{i,j} = \prod_{k=0}^{i-2}\left((j-1)\delta +n-k\right) ...
5
votes
0answers
67 views

A conjecture concerning the irreducibility of characteristic polynomials of Arndt matrices

Letting $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ binary matrix with ones along the main antidiagonal and everywhere below the main antidiagonal and ones along the antidiagonal two ...
2
votes
0answers
40 views

Computing a determinant

Suppose we have a $10 \times 10$ matrix $A$ which has $0$'s on the main diagonal (so that the trace of $A$ is $0$). Also suppose that $A^2=I$. How can we find a determinant of $A+2I$? Based on my ...
0
votes
2answers
34 views

Spectrum of a large matrix

How can I find the characteristic polynomial of the matrix $A$, so that I can find all of its eigenvalues, and hence the spectrum, so that I can use the spectrum to calculate the determinant of $A$? ...
11
votes
5answers
629 views

Defining the Determinant

The concept of determinant is quite unmotivational topic to introduce. Textbooks use such an "strung out" introductions like axiomatic definition, Laplace expansion, Leibniz'a permutation formula or ...
0
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0answers
22 views

How to form a matrix using an endomorphism?

Let $f\in \mathrm{End}_\mathbb{R}(\mathbb{R}[x]_{<4})$ be the endomorphism given by $x^2 \dfrac{\mathrm{d}^2}{\mathrm{d}x^2} + \dfrac{\mathrm{d}}{\mathrm{d}x} + 2\mathrm{i}d$. I'm supposed to be ...
0
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0answers
44 views

Is it true that det(A)=0 if $A^T=-A$ [duplicate]

I'm needing to find the determinant of a 3x3 matrix. I know that det(A)=det($A^T$) and $A^T =-A$. Am I right in thinking that this means det(A)=0 and if so why/why not?
2
votes
1answer
55 views

Sum of minors of matrix [closed]

If $ k $ is a diagonal minor of a matrix $ A \in M_{m \times n}(\mathbb{C}) $ then $ k $ has the following form: $$ M_{i_{1}, i_{2}, ..., i_{k}}^{i_{1}, i_{2}, ..., i_{k}}(A) = \begin{vmatrix} ...
1
vote
0answers
45 views

Some doubts regarding determinants

When can $\det(AB)$ be written as $\det(A) \det(B)$ and when not ? Is it possible that while we multiply two matrices $AB = C$ that determinant of $A$ and $B$ is not 0 but determinant of $C$ is $0$? ...
0
votes
0answers
11 views

Effective algorithmic calculation of gcd determinant

This is a contest problem taken from here: http://www.e-olymp.com/en/problems/3243 I need to calculate the following determinant: $$ D(1,\dots,n)=\begin{vmatrix} (1,1) & \cdots & (1,k) & ...
1
vote
0answers
31 views

Determinant and eigenvalues of an exponential matrix

So I have the following exponential matrix. $$e^{At} = \frac{1}{11}\begin{bmatrix}10+e^{11t} & -1+e^{11t} & -3+3e^{11t}\\-1+e^{11t} & 10+e^{11t} & -3+3e^{11t}\\ -3+3e^{11t} & ...
2
votes
3answers
64 views

Necessary and/or sufficient conditions for $A+B$ to be invertible

Let $A$ and $B$ be two $n\times n$ real invertible matrices. Are there necessary and/or sufficient conditions (involving only $A$ and $B$ separately, not $(A+B)$ iteself) for $A+B$ to be invertible? ...
3
votes
2answers
123 views

determinant of a very large matrix in MATLAB

I have a very large random matrix which its elements are either $0$ or $1$ randomly. The size of the matrix is $5000$, however when I want to calculate the determinant of the matrix, it is either ...
2
votes
1answer
21 views

in a non-full rank matrix, how to find the dependent rows especifically in MATLAB?

I have a very large matrix, which its determinant is zero, and hence it is not full rank. Now I wonder is there a way to find which rows are linearly dependent? specifically in MatLab is there any ...
1
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1answer
26 views

How to verify these determinant properties

I am confused about how to show that if $f:V \to V$ is a linear map, then the choice of basis is irrelevant when we compute det$(f)$. (where det f refers to computing the determinant of the matrix ...
0
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1answer
21 views

Nice form for determinant of a special matrix

I have a positive definite symmetric matrix that looks like where matrices $A,B,C,D$ are positive definite symmetric matrices. Is there a nice way to calculate the determinant? For example, the ...
1
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0answers
37 views

$\det({\matrix A - b \cdot\matrix I})$ simplification

I was wondering if there was a way to simplify the formula $\det({\matrix A - b \cdot \matrix I})$ where $\matrix A$ is a $4\times4$ matrix, $b$ an integer and $\matrix I$ the identity matrix.
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2answers
64 views

Square matrix whose determinant is already known

I was reading elementary algebra and suddenly I got stuck in a question which was Write a $3*3$ square matrix $A$ whose determinant is $75$ ? Now this can be done using hit and trial method or ...
2
votes
1answer
24 views

Invertibility of a certain matrix attached to a primitive root of unity.

Let $q\in \mathbb{C}$ be a primitive $n$th root of unity, for some $n>1$. Consider the $n^2\times n^2$-matrix $$M=\left( q^{ki+lj}\right)_{(k,l),(i,j)}$$ indexed by all pairs $(k,l), (i,j) \in ...
0
votes
3answers
94 views

How to calculate the determinant $\det(A+xI)$? [closed]

Let $A$ be an $n\times n$ symmetry matrix with the diagonal elements of $A$ are $0$ (the diagonal elements of $A$ are can also be any constant ). Let $I$ be an $n\times n$ identity matrix, and Let ...
1
vote
1answer
46 views

Symmetric matrix with integral entries and nonnegative determinant

I would like to propose a generalization of another question which I posed here yesterday. The main reason is the heuristic that if an inequality holds for the finite case, then an integral analogue ...
4
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2answers
69 views

Rank of a block matrix

Let $A$ and $B$ be two $n\times n$ matrices with real entries. Show that the matrix $M=\begin{pmatrix}A&I\\I&B\end{pmatrix}$ is of rank $n$ if and only if $A$ is invertible and $B=A^{-1}$ We ...
1
vote
1answer
64 views

Product matrix and induction

I am not sure which method to use here. Should I do it for $n=2$ and $n=3$ and then use induction on $n$? Let $\alpha_1,\alpha_2,\ldots,\alpha_n \in \mathbb{R}$, where $n \geq 2$. Show that ...
1
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2answers
61 views

How to prove matrix $A$ is invertible $\iff$ $\lambda=0$ is not an eigenvalue of $A$? [duplicate]

So we know that $A$ is not invertible $\iff$ det$(A)=0\iff \lambda=0$ is an eigenvalue of $A$. But the negation doesn't equal to the title statement. How would you prove the title question?
0
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1answer
44 views

Finding the determinant of a 4x4 matrix quickly

I know that I can find the determinant by working my way from 4x4 to 3x3 to 2x2 determinant. However that is long. Which other method can I use? \begin{bmatrix} 2 & 1 & 0 & 6 \\ 2 & ...
1
vote
1answer
58 views

Symmetric matrix with nonnegative determinant

Given positive reals $a,b$, and $c$, is it true that the matrix $$ \begin{pmatrix} a^4+b^4+c^4 & a^3+b^3+c^3 & a^2+b^2+c^2 \\ a^3+b^3+c^3 & a^2+b^2+c^2 & a+b+c \\ a^2+b^2+c^2 & ...
0
votes
1answer
44 views

Is there a way to “fill out” a $n\times n$ matrix so that the determinant is the same as an$ m\times m$ matrix with $m < n$?

Say I have a $3\times3$ matrix: \begin{bmatrix}x&y&1\\0&1&1\\1&1&1\end{bmatrix} The determinant is then $y-1$, is there a way to fill out a strictly larger matrix that still ...
1
vote
0answers
39 views

Permanent equals determinant

I would like to known if there is some characterization of matrices on which permanent equals determinant. I have quickly found out that if we set $perm(A) = \det(A)$, we arrive on $0 = \textrm{det ...
5
votes
1answer
51 views

Nonnegative determinant of a symmetric matrix

Consider the following matrix with nonnegative entries: $$ M=\begin{pmatrix} a & b & c & d \\ b & c & d & e \\ c & d & e & f \\ d & e & f & g \\ ...
2
votes
1answer
67 views

determinant of symmetric matrix with zeros in diagonal is even

Question is to prove that Determinant of a symmetric matrix of odd degree with integer entries and zeros in the diagonal is even.. For $A=(a_{ij})$ with $a_{ii}=0$ and $a_{ij}=a_{ji}$.. ...
5
votes
2answers
107 views

$Tr(A^2)=Tr(A^3)=Tr(A^4)$ then find $Tr(A)$

Let $A$ be a non singular $n\times n$ matrix with all eigenvalues real and $$Tr(A^2)=Tr(A^3)=Tr(A^4).$$Find $Tr(A)$. I considered $2\times 2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ ...
1
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2answers
66 views

Positive determinant of a symmetric matrix

Let $a,b,c,d,e$ positive reals for which $b<\sqrt{ac}$, $c<\sqrt{bd}$, and $d<\sqrt{ec}$. Then, consider the following (Hankel) matrix $$ M=\begin{pmatrix} a+4b+6c+4d+e & a+3b+3c+d & ...
2
votes
0answers
56 views

Maximal dimension of a vector space of square matrices in which every nonzero matrix is invertible

I'm interested in the maximal dimension of a subspace $V\leq\mathbb R^{n\times n}$ in which every nonzero matrix is invertible. Odd $n$: For odd $n$ the maximum is $1$: if $A$ and $B$ would be ...
1
vote
1answer
48 views

Diagonalizable matrices

Question is to prove that the set of all diagonalizable matrices are dense in $M_n(\mathbb{C})$. I am sure this question is discussed in this site previously but i am looking for a more constructive ...
6
votes
1answer
114 views

Problem of determinant when $A^{-1}+B^{-1}=(A+B)^{-1}$

I have two $4\times 4$ real matrices $A$ and $B$, and it is known that $A^{-1}+B^{-1}=(A+B)^{-1}$ ($A$, $B$ and $A+B$ are invertible). How can I prove that $\det (A)=\det (B)$?
10
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4answers
823 views

How to generalize the determinant as function

Hi I was asked to show that for any vector space $V$ over a field $\mathbb{F}$ of arbitrary dimension $n$ that if we fix some basis $\beta=\{w_1,\ldots,w_n\}$ that there is a unique function $D_\beta ...
0
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0answers
12 views

Proof of $det(A+B)=detA + <$Cof$A,B> + <A,$Cof$B> + detB$

For $A,B \in M^{3 \times 3}$, I want to prove the following property: $det(A+B)=detA + <$Cof$A,B> + <A,$Cof$B> + detB$ where $<.,.>$ denotes the usual inner product on ...
2
votes
1answer
23 views

Sufficient condition on rank of a matrix

Let $n\in \mathbb{N}$. Show that the determinant map $\det: M_n(\mathbb{R})\rightarrow \mathbb{R}$ is infinitely diffeentiable and compute the total derivative $d(\det)$ at every point of $A\in ...
2
votes
2answers
45 views

Rank $2$ matrices

Question is to show that the set of rank $2$ matrices in $M_{2\times 3}(\mathbb{R})$ is open... Let $M$ be the set of rank $2$ matrices in $M_{2\times 3}(\mathbb{R})$.. Let $N$ be the complement of ...
0
votes
0answers
37 views

Representing a matrix with unit determinant by the generators of the modular group?

I know that $a-b+c-1=1$ with $a,b$ and $c$ positive integers. Regrouping the terms we have $a-(b-c+1)=1$. I rewrite that equation as the determinant of a matrix, $$ e= \begin{pmatrix}{} a ...
2
votes
1answer
28 views

Linear relations among wedge products.

Let $v_1,v_2,v_3,v_4$ be $4$ vectors in a two dimensional space $V$. Then one can work out by hand that: $$(v_1\wedge v_2)(v_3\wedge v_4) + (v_1\wedge v_3)(v_4\wedge v_2) + (v_1\wedge v_4)(v_2\wedge ...
5
votes
2answers
75 views

Determinant of $(A+iB)(A-iB)$

Let $A,B$ real $n \times n$ matrices such that $A+iB$ is invertible. I want to prove (if true) that $|A+iB|\cdot|A-iB| >0$ Of course, this is the same as $\big| (A+iB)(A-iB) \big|>0$, so, ...
0
votes
1answer
26 views

How do linear dependent column vectors affect a rectangular determinant?

This is a problem from my signals and systems class. The question is asking for what values of b1, b2, b3 such that the system is controllable (determinant is 0). Image of question Is it even ...
0
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1answer
44 views

Proving the method of Tuncer ($4\times 4$ Determinant)

$\left( \begin{matrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\end{matrix} ...
3
votes
0answers
62 views

Calculate Determinant A size n

I am given homework like this, calculate the Matrix $$ \begin{bmatrix}x+1 ...
9
votes
3answers
1k views

I get a wrong determinant - why?

I'm trying to calculate the following determinant: $$\begin{vmatrix} a_0 & a_1 & a_2 & \dots & a_n \\ a_0 & x & a_2 & \dots & a_n \\ a_0 & a_1 & x & \dots ...